diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzkdsc" "b/data_all_eng_slimpj/shuffled/split2/finalzzkdsc"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzkdsc"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\n\n\nDespite over forty years of theoretical and observational studies, we still do not fully understand the emission mechanism responsible for the observed pulsar radiation. \n\n\nIn an attempt to fully understand the high energy nonthermal emission from pulsars, both direct and indirect methods have been used. Direct models attempt to describe fully the source physics that subsequently generates the radiation we see. Conversely, inverse models use the observed radiation in an attempt to infer information about the source physics and emission geometry itself. In this paper, we present the methodology for a generic inverse method, whose aim is to restrict pulsar geometric parameters ($\\alpha, \\chi$, and emission location), using few assumptions and many observational constraints. Inverse models have had a long history within pulsar astrophysics. The models effectively utilise the link between lightcurve morphology and local source geometry, to place restrictions on various source parameters, i.e., to restrict the pulsar's magnetic inclination ($\\alpha$), the observer's viewing angle ($\\chi$), and\/or to determine the emission location within the pulsar magnetosphere. Perhaps the best known of these models is the Rotating Vector Model (RVM) (\\cite{RC1969}), which constrains pulsar inclination and viewing angle. It does so by assuming that the {\\it radio} linear polarisation is described by a polarisation vector fixed to local B field lines, (located close to the magnetic pole), which sweeps by an observer's line of sight as the pulsar rotates.\n\nInverse methods in general, have evolved over time. Early ideas were to associate the width of pulse profiles with the relativistically beamed opening angle of isotropic radiation, from point sources corotating within the pulsars magnetosphere (e.g., \\cite{S1970}, \\cite{Z1971}, \\cite{ZS1972}). \\cite{M1983} proposed a model for pulsed optical $\\rightarrow$ $\\gamma$-ray emission based on relativistic electron beams producing highly beamed emission in the far (measured radially)  magnetosphere. Assuming emission from only the (purely dipolar) last open field lines of the {\\it equator} of an orthogonal rotator, the arrival phase of emitted photons (assumed tangential to the local B, subject to aberration) were mapped onto observer phase. Aligning this radius-to-phase mapping with the relative peak arrival phases for the Vela pulsar, Morini estimated that the radio, optical and $\\gamma$-ray emission for the Vela pulsar, originated at distances of $\\sim 0, 0.5$ and $0.7$ $R_{LC}$ from the neutron star surface. The advent of readily available computer power allowed the construction of radius-to-phase maps, as the assumption of tangential beaming means that the mapping is independent of the specific emission process. In the late 1980s, Smith and co-workers (\\cite{S1986}, \\cite{SJDP1988}) extended this idea somewhat, producing radius-to-phase maps for the whole equatorial {\\it plane} of an orthogonal {\\it retarded} dipole. RY95 (\\cite{RY1995}) succeeded in reproducing peak separations of the gamma ray pulsars, together with the radio-peak to gamma-ray peak relative phase offset, while simultaneously restricting pulsar inclination  and viewing angle. \\cite{DRH2004} describe how light flight time delays can be responsible for asymmetries in pulse profiles. DR03 \\citep{DR2003} developed a `Two Pole Caustic' model (TPC), which can produce double peaked (and to a more limited extent, single peak) light curves. \nWhile certain restrictive assumptions are sometimes necessary and justifiable, we have designed a general method tailored specifically at restricting unconstrained pulsar parameters. We effectively carry out a large scale computational search through pulsar parameter space to constrain pulsar parameters. The aim was to create a methodology of testing and refining hypotheses of how pulsars work. In this paper, we apply the model to pulsed optical emission from isolated neutron stars.\n\n\nFollowing a successful two years of operation, the Fermi Gamma-ray observatory has identified over fifty gamma-ray pulsars (\\cite{A2010b} \\& \\cite{SP2010}), of which 21 are radio-quiet gamma-ray pulsars and surprisingly, 11 are millisecond pulsars (\\cite{A2009}, \\cite{R2010}). From these observations, the polar-cap model has been effectively ruled out (\\cite{A2009}, \\cite{VHG2009}), although the polar cap will play a role in the initial particle acceleration. This leaves the slot-gap, outer-gap (extended outer gap for millisecond pulsars) and variations of the striped pulsar wind (\\cite{AS1979}, \\cite{CHR1986a}, \\cite{P2009} \\& \\cite{KSG2002}), as the main contenders for a theory of high-energy pulsar emission.\n\n\nIt has long been recognised that the polarisation of pulsar radiation gives an indication of the geometry of the emission zone. \\cite{TG2010} showed recently that the core and conal components of the radio emission from PSRs B1839+09, B1916+14 and B2111+46 were at low altitudes $<$ 5\\% of the light cylinder radius. \\cite{WW2009} mapped the magnetopshere of PSR B1055-52, indicating that the emission height was around $\\sim$700 km above the neutron star's surface. In a more general way, \\cite{TGG2010} have looked the geometry for radio emission assuming curvature radiation. Our objective in this paper, is to investigate the use of optical polarisation data to map the regions within the magnetosphere at which the pulsar emission can originate. The advantage of optical radiation is that its emission mechanism, incoherent synchrotron, has a simple relationship between its polarisation profile and the underlying geometry. It is also the only region where we have decent high-energy polarisation data, and such data is highly sensitive to the local emission physics conditions.\n\n\n\\section[]{Description of the Model}\n\nHistorically, inverse models have focused on restricting various geometrical parameters, such as pulsar inclination angle, viewing angle and\/or the specific emission site responsible for the observed emission. These variables are assumed unknown initially and define a parameter search space, where the aim is to restrict them by comparison with observation. However, all such models use rather restrictive initial conditions to constrain these parameters, for example, tangential beaming from specific subsections of the magnetosphere using predefined emissivity functions. \nBy removing these restrictive initial conditions and using modern compute power, we believe that the domain of applicability of inverse models can be increased. For example, we can apply the inverse approach to constrain and test emission properties, as well as geometric properties of pulsar high energy emission. Effectively, one considers the various emission parameters as being undefined and adds these parameters to the overall search space to be constrained. Extending inverse modelling in this way may provide a new way of constraining pulsar variables, and of approaching the pulsar problem in general. We describe in this section the overall approach we have designed and the specific inverse method we have developed to investigate pulsed nonthermal optical emission from pulsars.\n\n\nOur approach is composed of a number of conceptually distinct steps, outlined in the flowchart in Figure~\\ref{flow.diagram}. The main steps are subdivided into physical (P), computational (C), statistical (S), and mapping (M) elements. Briefly, these steps constitute: (1) the application of a plausible physical model responsible for the emission, (2) the computational implementation of this model, resulting in the creation of phase resolved lightcurves ($\\xi(\\Phi,\\alpha,\\chi)$)\\footnote{Emission from a given pulsar is simulated and phase resolved ($\\Phi$) lightcurves produced for a range of viewing angles ($\\chi$). This process is repeated using different inclinations ($\\alpha$) for the magnetic axis (which is not in general constrained from observations). $\\xi$ can represent numerous observational properties, in this case the set of Stokes parameters I,Q,U,V.}, (3) the statistical comparison of the simulated lightcurves to observations, which results in a restriction of $(\\alpha,\\chi)$ parameter space enabling, (4) an `inverse mapping' into the magnetosphere, locating the regions of emission responsible the best fitting lightcurves in the first place. \\\\\n\\begin{figure}\n\\centering\n{\\includegraphics[scale=0.4]{flow_inverse.eps}}\n\\caption[Representation of `Inverse Mapping'\/`Search Algorithm' approach in flow diagram form]{\nRepresentation of `Inverse Mapping'\/`Search Algorithm' approach in flow diagram form. The approach\nis conceptually divided into physical, computational, statistical and mapping\/inverse mapping components.}\n\\label{flow.diagram} \n\\end{figure}\n\nPhase resolved Stokes parameters are produced for all possible values of $0\\deg \\le \\alpha \\le 180\\deg$ and $0\\deg \\le \\chi \\le 180\\deg$, in finite increments $\\Delta \\alpha$ and $\\Delta \\chi$, based on the assumption that a simple radiative emission process is occurring globally within the magnetosphere. The computational model divides the magnetosphere into a grid of points and the phase resolved angular distribution of photons from each point ($\\bm{R}$) is recorded as $\\xi(\\bm{R},\\Phi,\\alpha,\\chi)$. The summation of $\\xi$ from all grid points creates a mapping from magnetospheric location to phase resolved observer profiles (as a function of $(\\alpha,\\chi)$). The statistical component selects the $\\xi(\\alpha,\\chi)$ which best fits the observations and $\\xi(\\bm{R},\\Phi,\\alpha,\\chi)$ allows an `inverse' mapping of the photon paths back into the magnetosphere.\\\\\n\n\nThese steps constitute what we refer to as the `Inverse mapping' or search algorithm approach to isolating the regions within the magnetosphere which may be responsible for pulsed optical emission. The main steps in the process are as follows: \\\\\n\n\\newcounter{listno}\n\\begin{list}{\\bfseries\\upshape P--\\Roman{listno}}\n {\\usecounter{listno}\n  \\setlength{\\rightmargin}{0.01\\columnwidth}\n  \\setlength{\\leftmargin}{0.07\\columnwidth}}\n\n\\item \nParticle motion (outside of accelerated regions) is dominated by the magnetic field, making the field an important element in any model $\\Rightarrow$ we propose a magnetic field structure in the form of a retarded dipole, as the standard Deutsch formalism, \\cite{ML1999}. \n\n\\item It is plausible that synchrotron radiation is responsible for pulsed optical emission, with the observed power law spectral form indicating that the underlying particle spectrum is also of this form. Using the premise of first order approximations, together with no a-priori assumptions regarding favourable emission locations, we assume that each point in the magnetosphere can emit synchrotron radiation from an underlying particle power law, where the total number density of particles at any one point is equivalent to the Goldreich-Julian density ($n_{GJ}\\propto \\Omega \\cdot B_z(r)$).\n\n\n\\item Each point $\\bm{R}$ is allowed to radiate synchrotron radiation, which is characterised by the properties of the particle population at that point - essentially the energy distribution of the particles (a power law from {\\bf P-II} above), and the pitch angle distribution (hereafter PAD) of the particles. The pitch angle distribution is perhaps the least well known parameter of pulsar emission models. Since the PAD is not well constrained, we assume various forms for the PAD and analyse the resultant lightcurves to choose which PADs may best represent the emission. For all PADs, we assume that the emitting particles are symmetrically distributed about the magnetic axis, with a cutoff occurring at a specific pitch angle, beyond which no emission occurs, {\\small PAD}$_{co}$.\n\\end{list}\n\n\\newcounter{listno1}\n\\begin{list}{\\bfseries\\upshape C--\\Roman{listno1}}\n{\\usecounter{listno1}\n  \\setlength{\\rightmargin}{0.01\\columnwidth}\n  \\setlength{\\leftmargin}{0.07\\columnwidth}}\n\n\\item Computationally, we represent the 3D volume of the magnetosphere as a grid. Each point on the grid is specified by local parameters such as $(\\bm{R},\\bm{B},n_{GJ},\\bm{v}_{co})$, where $\\bm{R}$, $\\bm{B}$, $n_{GJ}$, and $\\bm{v}_{co}$ are the spatial coordinate, the magnetic field, the local particle density and the corotational velocity respectively, at that point. The grid itself, is based on a spherical geometry, consisting of a series of fixed concentric spheres. The origin of the spheres is located at the centre of the neutron star, with points located such that $\\bm{R}=(R,\\theta,\\phi)$ in usual spherical coordinates.\n\n\\item At each grid point ($\\bm{R}$), a power law distribution of particles, having a specific PAD, emits synchrotron radiation into a certain section of sky of solid angle $d\\Omega$. Relativistic aberration, beaming and doppler shifts will alter the direction of emission, the irradiated solid angle and frequency\nof radiation, as seen by a sky based observer compared to an observer in the particle rest frame. These effects are considered and the emission from each point $\\bm{R}$, as seen by a sky based observer, is recorded in increments of $\\Delta \\chi = 1\\deg$, for the range of $0\\deg \\le \\chi \\le 180\\deg$.\n\n\\item Since one aim of this approach is to restrict both the inclination angle $\\alpha$ {\\it and} the viewing angle $\\chi$, step {\\bf C-II} is repeated for all inclination angles with $0\\deg \\le \\alpha \\le 180\\deg$, in discrete increments of $\\Delta \\alpha$.\n\\end{list}\n\n\\newcounter{listno2}\n\\begin{list}{\\bfseries\\upshape S--\\Roman{listno2}}\n {\\usecounter{listno2}\n  \\setlength{\\rightmargin}{0.01\\columnwidth}\n  \\setlength{\\leftmargin}{0.07\\columnwidth}}\n\n\\item Steps {\\bf C-II} and {\\bf C-III} create a phase space $(\\alpha,\\chi)$, where each point in this phase space contains phase resolved lightcurves of the form:\n\\begin{equation}\n\\xi(\\bm{R}, \\Phi) = \\xi(R,\\theta,\\phi,\\Phi),\n\\end{equation}\nwith $0\\deg \\le \\chi \\le 180\\deg$, $0\\deg \\le \\alpha \\le 180\\deg$, and where  $\\xi$ represents each of the four Stokes parameters $(I,Q,U,V)$. A $\\chi$-squared goodness of fit test is performed between each of the simulated Stokes parameters and the observed Stokes parameters, at each point in $(\\alpha,\\chi)$ phase space, to produce the best fitting $(\\alpha,\\chi)$ combination.\n\\end{list}\n\n\\newcounter{listno3}\n\\begin{list}{\\bfseries\\upshape M--\\Roman{listno3}}\n {\\usecounter{listno3}\n  \\setlength{\\rightmargin}{0.01\\columnwidth}\n  \\setlength{\\leftmargin}{0.07\\columnwidth}}\n\n\\item Knowing the best fitting $(\\alpha,\\chi)$ combination, $\\xi(R,\\theta,\\phi,\\Phi,\\alpha,\\chi)$ is used to inverse map the constituent photons back into the magnetosphere, thereby locating the origin of the emission, which creates the best fitting lightcurves ($\\xi$). The visualisation of these regions will hopefully yield an indication of the originating locations.\n\\end{list}\n\nIn the following sections, each of these steps in illustrated in more detail, applying the search algorithm to a simulated pulsar with Crab like parameters: ($P=33$ ms, $\\dot{P}=4.209\\times10^{-13}\\rm{ss^{-1}}$ and $B_{\\rm{surf}}=3.8 \\times 10^{12}$ G).\n\n\\section[]{Description of the Physical Model}\n\nIt is assumed in this modelling that at any point in the magnetosphere, radiation arises through synchrotron radiation from a power law spectrum of particles: the particle index, p, is fixed via the observed photon spectral index s, through the relation, s=(p+1)\/2, where:\n\n\\begin{equation}\nF_\\nu \\propto \\nu^{-s} \\quad \\Rightarrow \\quad N(E) \\propto E^{-p}.\n\\end{equation}\n\nWhen simulating emission over a range of frequencies (e.\\,g., for U, B, V bands), contributions are summed only from those particles whose energies, are such that they contribute significantly to emission at the {\\it observed} frequency $\\nu$.\n\n\\subsection{Synchrotron Radiation}\n\nSynchrotron radiation from a single particle, spiralling around a local magnetic field ($\\bmath B$), at a pitch angle $\\alpha$ (where $\\hat{n} \\cdot \\hat{B} = \\cos \\alpha$ with $\\hat{n}$ the instantaneous velocity of the particle), produces a continuum spectrum of frequencies with a peak emissivity close to a critical frequency, $f_c$ where:\n\n\\begin{equation}\nf_c(E,B,\\theta) = \\frac{3}{2} f_B \\sin \\theta \\left( \\frac{E}{E_o} \\right)^2,\n\\label{fc}\n\\end{equation}\n\nwith $f_B=e B\/2\\pi m$, being the particle cyclotron frequency. To first order, the spectrum exhibits a quadratic rise and exponential drop as the frequency passes through the critical value. At lower frequencies, the spectrum consists of discrete harmonics of the fundamental (equation~\\ref{fundamental_freq}) frequency below which no emission is possible.\n\n\\begin{equation}\nf_f = \\frac{f_B}{\\sin^2 \\theta} \\frac{E_o}{E}  \\label{fundamental_freq}.\n\\end{equation}\n\nThe emitted radiation is in general, elliptically polarised with the principle axes of the polarisation ellipse aligned parallel and perpendicular to the projection of $\\bmath{B}$ on the plane transverse to the emission direction, $\\hat{l}$ (see Figure~\\ref{particle_geometry}). The angular dependence of the polarised emissivity depends strongly on the angle between the line of sight ($\\hat{l}$) and $\\hat{n}$, where we write, $\\hat{n} \\cdot \\hat{l} = \\cos \\psi$ with $sgn(\\psi)=\\alpha-\\theta$. The case $\\psi = 0$, results in linear polarisation perpendicular to $\\bmath{B}$, $\\psi > 0$ denotes elliptical polarisation having the major axis perpendicular to the projection of $\\bmath{B}$, and $\\psi < 0$ indicates right handed polarisation and $\\psi > 0$ left handed.\n\n\\begin{figure}\n\\center\n{\\includegraphics[scale=0.4]{emission_source_geometry-1.eps}} \n\\caption{Important source and observer geometric parameters relating to synchrotron emission.}\n\\label{particle_geometry}\n\\end{figure}\n\nFor a population of particles, one must integrate emission over the velocity distribution, which amounts to an integration over the pitch angle distribution and over the range of particle energies present. Results for a power law spectrum of particles ($N(E) \\propto E^{-\\gamma}$) are well known; the emitted radiation spectrum is also a power law, with the photon spectral index related to the particle index, where $\\alpha=(\\gamma+1)\/2$ and $\\gamma < 1\/3$, in order to maintain a finite integral. This monotonic spectrum is of course a special case, which results from integrating over a population having infinitely wide energy bounds. So that practically, the formalism can be applied only at frequencies which are unaffected by any issues related to the nature of finite particle energy bounds. \n\n\\subsection{A Truncated Power Law Spectrum of Particles}\n\nIn any real situation, one is dealing with particles having finite upper and lower energy bounds, which alter the spectrum significantly from the classical monotonic scenario. For the purposes of this work, the angle dependant and integrated polarisation properties of synchrotron emission from a truncated power law spectrum of particles, with $N(E) \\propto E^{-\\gamma}$ is considered. In this spectrum, any given particle's energy is confined between lower ($E_1$) and upper ($E_2$) limits, such that $E_1 \\le E \\le E_2$ having a (potentially) isotropic, axially symmetric (about $\\bm{B}$) pitch angle distribution. Such a distribution is rather general and has been dealt with in detail by GLW74 \\citep{GLW1974}. They derive emissivity properties using the polarisation tensor for light, which represents the cross correlated quadratic components of the electromagnetic wave field as a rank 2 tensor, $\\rho_{\\alpha\\beta}$ (where $\\alpha$ and $\\beta$ represent the component directions in the transverse plane). The tensor $\\rho_{\\alpha\\beta}$, contains the complete polarisation content of the radiation and can therefore be related to the more commonly used Stokes parameters, as illustrated in equation~\\ref{polarisation_tensor}. For reference purposes, the underlying polarised emissivity properties are given here as equations~\\ref{stokes}, reproduced from GLW74.\n\n\\begin{equation}\n\\left.\n\\parbox{0.5\\textwidth}{\n\\begin{eqnarray*}\nI & = & \\frac{A\\mu e^2c}{2\\sqrt{2}}(\\frac{3}{2})^{\\gamma\/2}\\phi(\\theta)({\\nu}_H \\rm{sin} \\theta)^{(\\gamma+1)\/2} \\\\\n \t                                   &  &\\cdot\\:\\: \\nu^{-(\\gamma-1)\/2} \\left[ \\mathcal{J}_{(\\gamma+1)\/2} \\right]_{x_2}^{x_1} \\nonumber  \\\\ \nQ & = & \\frac{A\\mu e^2c}{2\\sqrt{2}}(\\frac{3}{2})^{\\gamma\/2}\\phi(\\theta)\n            ({\\nu}_H \\rm{sin}\\theta)^{(\\gamma+1)\/2} \\\\\n            &  & \\cdot\\:\\: \\nu^{-(\\gamma-1)\/2}\\left[ \\mathcal{L}_{(\\gamma+1)\/2} \\right]_{x_2}^{x_1}  \\nonumber \\\\\nU & = & 0 \\nonumber  \\\\\nV & = & \\frac{A\\mu e^2c}{\\sqrt{3}}(\\frac{3}{2})^{\\gamma\/2}\\phi(\\theta)\n                            \\rm{cot} \\theta ({\\nu}_H \\rm{sin}\\theta)^{\\gamma\/2+1}\n                             \\nu^{-(\\gamma\/2)} \\nonumber  \\\\\n                           &  & \\times \\left[\\mathcal{R}_{(\\gamma\/2+1)}+\n                                 (1+g(\\theta))(\\mathcal{L}_{\\gamma\/2} -\\frac{1}{2}\\mathcal{J}_{\\gamma\/2}) \n                                 \\right]_{x_2}^{x_1} \\label{stokes}\n\\end{eqnarray*}}\n\\right.       \n\\end{equation}\nwhere:\n\\begin{equation}\n\\mathcal{J}_{n}(x)=\\int_o^x \\xi^{n-2} F(\\xi) d\\xi, \\hs \\hs \\mathcal{L}_{n}(x)=\\int_o^x \\xi^{n-2} F_p(\\xi) d\\xi\n\\end{equation}\n\\begin{equation}\n\\mathcal{R}_{n}(x)=\\int_o^x \\xi^{n-2} F_s(\\xi) d\\xi, \\hs \\hs F(\\xi)= \\int_\\xi^\\infty K_{5\/3}(y)dy\n\\end{equation}\n\\begin{equation}\nF_p(\\xi)= \\xi K_{2\/3}(\\xi), \\hs \\hs F_s(\\xi)= \\xi K_{1\/3}(\\xi)\n\\end{equation}\nso that:\n\\begin{equation}\n\\mathcal{J}_{n} \\equiv \\int_o^\\infty \\xi^{n-2} F(\\xi) d\\xi  \\hs \\rm{etc.,}\n\\end{equation}\nwhere the $K_\\alpha$, are modified Bessel functions and the variable $x \\equiv f\/f_c$, where $f_c$ is the critical frequency given by \\ref{fc}:\n\nThe effect of the $truncated$ power law energy spectrum is seen in the fact that the functions modifying the observed Stokes parameters (i.e., $\\mathcal{J}_n(x), \\mathcal{L}_n(x)$ and $\\mathcal{R}_n(x)$), must be evaluated between upper ($x_1$) and lower ($x_2$) limits of the parameter `x', where $x=x(B,\\theta,E)$. Each polarised emissivity parameter is dependant on a functional combination of $\\phi(\\theta), f_B \\sin \\theta, f, \\gamma$, and this functionality is modulated by finite integrals over the particle population (having integrand `x').\n\n\nAs analysed in detail by GLW, having a finite particle energy range results in an emission spectrum with lower ($f_l$) and upper ($f_u$) frequency bounds. This spectrum possesses roughly two spectral breaks at frequencies $f_a$ and $f^{(i)}_b$, where $f_{lo}<f_a<f^{(i)}_b<f_{hi}$, with differing spectral indices for each polarisation parameter, each asymptotically dependent on the power law index $\\gamma$ (see for example, Figure~\\ref{spectrum_with_corrxn_factors}).          \\\\\n\nThe frequency bounds, the internal transition frequencies and the form of the spectral variation, are all the result of energy cut-off effects. All effects have a simple physical basis and can be described in terms of  the very useful variable of integration `x', which contains the frequency of interest, f, and encodes all relevant physical factors relating to the emission process, ($E, {\\bf B}, \\theta$). In fact, the fiducial frequencies are derived from the corresponding fiducial `x-factors', designated $x_l<x_a<x^{(i)}_b<x_u$.\n\nPhysically, both the low and high energy frequency cutoffs are due to the absence of particles with energies, $E$, greater than the upper limit $E_2$ ($f_{lo} \\propto E_2^{-1}$, $f_c \\propto E_2^2$). Above $x^{(i)}_b$, we have $x_1 \\ll x \\ll x_2$, so that the upper and lower limits of integration can be approximated to $\\infty$ and 0 respectively. This corresponds to the classical scenario in which $f_{c1} \\ll f \\ll f_{c2}$; the case in which the integrals are dependant solely on the spectral index $\\gamma$, and can be expressed in terms of a combination of Euler Gamma functions. The photon indices for the stokes parameters are now the classical values, the frequency dependencies already given in relations \\ref{stokes}, i.e., $\\alpha=(\\gamma-1)\/2$ for Stokes I and Q, and $\\alpha=\\gamma\/2$ for Stokes V. Below $x^{(i)}_b$, the finite limits of $x_1$ and $x_2$ cannot be ignored, and one gets a `falling away' from the classical photon index as the frequency decreases. This results in an explicit dependence on $x_1$ and $x_2$ (and subsequently an altered frequency dependence). Deriving Power Law Expansion (PLE) approximations for the relevant integral evaluations, GLW74 showed that the asymptotic spectrum should now vary as $(f\/f_B \\sin \\theta)^{1\/3} E^{\\gamma-1\/3}$ for Stokes I and Q, and as $E^{-\\gamma}$ for Stokes V (where $E=E_1$ or $E_2$).\n\n\\begin{figure}\n\\center\n{\\includegraphics[scale=0.4]{glw_eta2.eps}} \n\\caption{Spectral variation of emission (Stokes I) from a truncated power law energy spectrum. The effect that decreasing the local magnetic field strength ($\\bm{B}$) and increasing the upper power law cutoff energy ($E_2$) have on the spectral break points is indicated by the directional dotted arrows at the appropriate points. Our spectral range covers UBV.\n}\n\\label{spectrum_with_corrxn_factors}\n\\end{figure}\n\n\nAt $f^{(i)}_b$, again, the finite limits of $x_1$ and $x_2$ cannot be ignored and one gets a `falling away' from the classical photon index as frequency decreases. Power law approximations to the integrals in relations \\ref{stokes}, show an asymptotic dependence on $x_1$, providing a spectral variation of $f^{1\/3}$.\n\nAs lower energy particles will have a higher fundamental frequency than higher energy particles, a point may come where some of the lower energy particles in the population cannot contribute to the frequency of interest, and the frequency at which this occurs, is designated $f_a$. GLW74 deal with this case by determining the lowest energy particles from the population which can contribute to emission at frequency f (designated by $E_1^{'}$ where $E_1^{'} > E_1$), and replacing the upper limit of integration with $x_1^{'}$, where $x_1^{'}=f\/f_c(E_1^{'})$. It is this replacement, which results in the augmented spectral index of $\\alpha=-\\gamma$ at frequencies below $f_a$.\n\nThe different energy cut-off effects delineate different regions in which different physical effects alter the emission characteristics. The effects are encoded in the integration x-factors, both in the form of $x_l$ and the extent of the range $(x_l,x_2)$, over which emission is allowed. Figure~\\ref{spectrum_with_corrxn_factors} shows the expected form of the emissivity for a truncated power law particle population emitting synchrotron radiation.\\footnote{We note that the photon indices quoted above are asymptotic values evaluated in the limit of `small', using power law expansion representations of the functions $\\mathcal{J}_{n}(x), \\mathcal{L}_{n}(x)$ and $\\mathcal{R}_{n}(x)$ derived by GLW74. The transition frequencies and photon indices are therefore asymptotic and indicative of the actual spectrum, which should transit smoothly through the transition frequencies and deviate smoothly from the asymptotic spectral indices.}\n\nIn our approach to modelling emission from a truncated power law particle population, we follow the GLW74 approach and describe the spectral variability in terms of correction factors, $C^i$, $i=1,2,4$ applied to the standard power law dependences (see Appendix \\ref{appendix_1}). We note that in limited circumstances, it is possible for $x_a>x^{(i)}_b$, so that the spectral emissivity has a lower and upper bound with a single internal transition frequency. This situation would arise when the upper and lower bounds are sufficiently close together. For completeness, we include the relevant correction factors in Appendix \\ref{appendix_1} (as this situation was not discussed in GLW74).\n\n\n\\subsection{Effect of source motion}\n\nThe description of the polarisation parameters above is valid when the magnetic field is stationary relative to an observer. In the case of emission from synchrotron radiating particles constrained to move along magnetic field lines in a pulsar magnetosphere, the underlying field structure is (relativistically) rotating. To describe the effect of the source motion on the observed polarised emission, we consider how a general Lorentz boost transforms the radiation fields. Also, to extract angular dependences and for frame transformations, it is convenient to use the polarisation tensor, $\\rho_{\\alpha\\beta}$, representation of light.   \n\nIn our notation, a noninertial observer  in the corotating frame $S$ (basis $O_{xyz}$), sees a stationary magnetic field $\\bm{B}$ and views this field along a direction $\\bm{\\hat{n}}$, such that $\\bm{B}_f \\cdot \\bm{\\hat{n}} \\propto  \\cos \\psi$. The observer sees light from a particle, if $\\bm{\\hat{n}}$ is sufficiently close to $\\bm{\\tau}(t)$, where $\\bm{\\tau}$ describes the trajectory of the charged particle (see Figure~\\ref{particle_geometry}). The electromagnetic field of the light oscillates in a plane transverse to $\\bm{\\hat{n}}$ (the `observer plane' K), and may be expressed in component form along two mutually perpendicular axes $(\\bm{\\hat{i}}_1, \\bm{\\hat{i}}_2)$ in the plane K. Frame $S'$ observes the magnetic field $\\bm{B}_f$ to move at an arbitrary constant velocity $\\bm{\\beta}=\\bm{v}\/c$. An observer in $S'$ will therefore see light emitted at an aberrated direction $\\bm{\\hat{n}}'$ about the boosted field direction $\\bm{B}'$. The components of the electromagnetic wave will also be boosted ($\\bm{E}',\\bm{B}'$). A general Lorentz boost, where frame $S^{\\prime}$ moves at an arbitrary constant velocity $\\bm{\\beta}$ to frame $S$, is given as $\\Lambda^\\mu_\\nu$ (equation~\\ref{lorentzboost}) where a four-vector $\\vec{x}$ transforms as ${x'}^\\mu = {\\Lambda^{\\mu}}_\\nu x^\\nu$.\n\n\\begin{equation}\n\\label{lorentzboost}\n\\Lambda^\\mu_\\nu = \\left[ \\begin{array}{llll}\n  \\gamma        & -\\beta_x\\gamma  & -\\beta_y\\gamma & -\\beta_z\\gamma \\\\\n -\\gamma\\beta_x & 1+\\frac{\\gamma-1}{\\beta^2}{\\beta_x}^2 & \\frac{\\gamma-1}{\\beta^2}\\beta_x\\beta_y & \\frac{\\gamma-1}{\\beta^2}\\beta_z\\beta_x \\\\\n -\\gamma\\beta_y & \\frac{\\gamma-1}{\\beta^2}\\beta_x\\beta_y & 1+\\frac{\\gamma-1}{\\beta^2}{\\beta_y}^2 & \\frac{\\gamma-1}{\\beta^2}\\beta_z\\beta_y \\\\\n -\\gamma\\beta_z & \\frac{\\gamma-1}{\\beta^2}\\beta_x\\beta_z & \\frac{\\gamma-1}{\\beta^2}\\beta_y\\beta_z & 1+\\frac{\\gamma-1}{\\beta^2}{\\beta_z}^2\\\\\n\\end{array} \\right] \n\\end{equation}\n\n\n\\noindent The electric and magnetic fields, transform as the components of the antisymmetric tensor $F_{\\mu\\nu}$, (equation~\\ref{antisym_eb}), so that ${F'}_{ik}={\\Lambda^{\\beta}}_k {\\Lambda^{\\alpha}}_iF^{\\alpha \\beta}$. The transformed wave four-vector ($\\bm{k'}$), electric ($\\bm{E'}$) and magnetic ($\\bm{B'}$) fields are given in equations~\\ref{E_trans} $-$ \\ref{k_trans}. \n\n\\begin{equation}\nF_{\\mu\\nu} = \\left[ \\begin{array}{cccc} \n      0   & -E_x & -E_y & -E_z \\\\\n      E_x & 0    & B_z  & -B_y \\\\\n      E_y & -B_z & 0    & B_x \\\\\n      E_z &  B_y & B_x  & 0   \n      \\label{antisym_eb}\n      \\end{array} \\right]\n\\end{equation} \n     \n\\begin{eqnarray}\n\\bm{E}' & = & \\gamma(\\bm{E} + \\left[ \\bm{\\beta H} \\right]) \n       - \\frac {\\gamma^2}{(\\gamma+1)}\\bm{\\beta} (\\bm{\\beta E}) \\label{E_trans}\\\\\n\\bm{B}' & = & \\gamma(\\bm{B} + \\left[ \\bm{\\beta B} \\right]) \n       - \\frac{\\gamma^2}{(\\gamma+1)}\\bm{\\beta} (\\bm{\\beta B}) \\label{B_trans}\n\\end{eqnarray}\n\n\\begin{equation}\n{k'}_i = \\frac{\\omega}{c}\\gamma(1-\\bm{\\beta}\\bm{n})\n         \\left\\{ \n         1,\\left( \\frac{\\bm{\\beta-\\hat{n}}}{1-(\\bm{\\beta}\\bm{n})}-\n                  \\frac{1}{(\\gamma+1)}\\frac{\\bm{\\beta}(\\bm{\\beta n})}{1-(\\bm{\\beta}\\bm{n})}\n           \\right) \\right\\} \\nonumber\n\\end{equation}\n\n\\begin{equation}\n= \\left( \\frac{{\\omega}'}{c},\\frac{-{\\omega}'}{c}\\bm{\\hat{n}'} \\right) \\label{k_trans}\n\\end{equation}\n\n\nIt is now necessary to describe the observed intensity and polarisation in terms of quantities seen in the observers frame (S'). For this, the polarisation tensor representation ($\\rho_{\\alpha\\beta}$) of light is most useful, given by:\n\\begin{equation}\n\\label{polarisation_tensor1}\n\\rho_{\\alpha \\beta} = \\frac{cR^2}{4\\pi}\\fte_{\\,\\,\\alpha} {\\fte}^{\\,\\,\\,*}_{\\,\\,\\beta},\n\\end{equation}\n\\noindent where $\\fte_{\\,\\,\\alpha}$ and $\\fte_{\\,\\,\\beta}$ are two orthogonal components of the electromagnetic wave field. The polarisation tensor can also be expressed in terms of the more widely used Stokes parameters, as in equation~\\ref{polarisation_tensor}.\n\n\\begin{equation}\n\\label{polarisation_tensor}\n\\rho = \\left[ \\begin{array}{cc} \n      \\rho_{11} & \\rho_{12} \\\\\n      \\rho_{21} & \\rho_{22} \\\\\n      \\end{array} \\right] \\propto \n      \\left[ \\begin{array}{cc} \n      \\frac{1}{2}(I+Q) & \\frac{1}{2}(U-iV) \\\\\n      \\frac{1}{2}(U+iC) & \\frac{1}{2}(I-Q) \\\\\n      \\end{array} \\right]  \n\\end{equation}\n\n\n\\begin{figure*}\n\\center\n{\\includegraphics[scale=0.6]{nonrotating.eps}\n\\hfill\n\\includegraphics[scale=0.6]{rotating.eps}} \n\\caption{Effect of a relativistic source motion on the observed Stokes parameters, determined here for $B=1.9 \\times 10^{6}$ G, $\\beta=0.87$.}\n\\label{boosting_on_stokes}\n\\end{figure*}\n\nSince the fields of the wave remain transverse in any frame, it is apparent that the tensor $\\rho_{\\alpha\\beta}$ will remain two dimensional in any new frame. Also, because the transformation is real, the real and imaginary parts of the tensor transform independently. These properties allow us to boost and rotate the electric field components from plane K in frame S, to plane K' in frame S'. Before boosting $\\rho_{\\alpha\\beta}$, any reference to $\\bm{B}$ in the expression for $\\bm{E'}$ is removed, using the substitution $\\bm{B}=[\\uv{n}\\rm{E}]$. Also, a right handed orthonormal basis set (denoted $\\rm{O}_{j_1j_2j_3}$) is defined in the observers (S') plane (K'). The projection of the pulsars rotation axis ($\\bm{\\Omega}$) on the plane K' is chosen as a uniform reference direction $\\uv{j_1}$, so that the basis set designated as $\\rm{O}_{j_1j_2j_3}$, can be defined through equation~\\ref{j_basis}.\n\n\\begin{equation}\n\\bm{j}_1=\\frac{[\\bm{n}'\\uv{\\Omega}]}{\\mid[\\bm{n}'\\uv{\\Omega}]\\mid} ; \\quad \\bm{j}_2=[\\bm{n}'\\bm{j}_1]\n               ; \\quad \\bm{j}_3=\\bm{n}' \\label{j_basis}\n\\end{equation}\n\n$\\uv{j_1}$ and $\\uv{j_2}$ are orthogonal and in the observers plane (K'), perpendicular to the boosted emission direction $\\uv{j_3}$. We now express  $\\bm{E'}  =  \\bm{E_1}'\\uv{j_1} + \\bm{E_2}'\\uv{j_2}$, where $\\bm{E_1}' = (\\bm{E}' \\uv{j_1})$ and  $\\bm{E_2}' = (\\bm{E}' \\uv{j_2})$. Since $\\bm{E}= iE_1 \\uv{i_1} + E_2 \\uv{i_2}$, it is possible to write $\\bm{E}'$ in the $\\rm{O}_{j_1j_2}$ basis explicitly in terms of $\\bm{E_1}$ and $\\bm{E_2}$ as follows:\n\n\\begin{eqnarray}\n({E}_1'\\bm{j}_1, \\, {E}_2'\\bm{j}_2) & = & (\\{iE_1x_{11}+E_2x_{21}\\}\\bm{j}_1, \\, \\nonumber \\\\\n  &  & \\{iE_1x_{12}+E_2x_{22}\\}\\bm{j}_2),\n\\label{transformed}\n\\end{eqnarray}\n\nwhere the x factors ($x_{11}$ etc.), are given by equation~\\ref{transformation_factors}, and are dot products between the $\\rm{O}_{j_1j_2}$ basis vectors and the $E_1$ and $E_2$ components of $\\bm{E}'$.\n\n\\begin{eqnarray}\n  x_{\\alpha\\beta}=\\frac{1}{(1+(\\bm{\\beta}\\bm{n}'))}\\Bigg\\{ \\frac{\\bm{l}_\\alpha\\bm{j}_\\beta}{\\gamma}+ \\nonumber \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\\\\n  \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:(\\bm{\\beta}\\bm{l}_\\alpha)\\left[ (\\bm{n}'\\bm{j}_\\beta) +\\gamma(\\bm{\\beta}\\bm{j}_\\beta)\\left( 1+\\frac{\\gamma}{(\\gamma+1)}(\\bm{\\beta}\\bm{n}') \\right) \\right] \\Bigg\\} \\label{transformation_factors}\n\\end{eqnarray}\n\nEquation~\\ref{transformed} expresses how the components of $\\bm{E}$ and $\\bm{B}$ in plane K, written in terms of the basis ($\\bm{\\hat{i}}_1, \\bm{\\hat{i}}2$), transform into components of $\\bm{E}'$ and $\\bm{B}'$ in plane $K'$, expressed along a basis $\\bm{\\hat{j}}_1, \\bm{\\hat{j}}_2$. Using equation~\\ref{polarisation_tensor}, the components of the emission-polarisation tensor ${\\rho'}_{\\alpha \\beta}$ in frame $S'$ can be determined as follows:\n\n\\begin{equation}\n \\begin{array}{lll} \n\\rho_{11} = \\fte_{\\,\\,\\,1}{\\fte}^{\\,\\,\\,\\,*}_{\\,\\,\\,1}x_{11}x^{*}_{22} + \\fte_{\\,\\,\\,2}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,2}x_{21}x^*_{21}, \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\\\\n\\\\\n\\rho_{12} = \\fte_{\\,\\,\\,2}{\\fte}^{\\,\\,\\,\\,*}_{\\,\\,\\,2}x_{21}x^*_{22} + \\fte_{\\,\\,\\,1}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,1}x_{11}x^*_{12} \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\\\\n\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: +i(\\fte_{\\,\\,\\,1}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,2}x_{11}x^*_{22} - \\fte_{\\,\\,\\,2}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,1}x_{21}x^*_{12}), \\\\\n\\\\\n\\rho_{21} = \\fte_{\\,\\,\\,2}{\\fte}^{\\,\\,\\,\\,*}_{\\,\\,\\,2}x^*_{21}x_{22} + \\fte_{\\,\\,\\,1}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,1}x^*_{11}x_{12} \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\\\\n\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: +i(\\fte_{\\,\\,\\,1}^{\\,\\,\\,*}{\\fte}_{\\,\\,\\,2}x_{11}^*x_{22} - \\fte_{\\,\\,\\,2}^{\\,\\,\\,*}{\\fte}_{\\,\\,\\,1}x^*_{21}x_{12}), \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\\\\n\\\\\n\\rho_{22} = \\fte_{\\,\\,\\,2}{\\fte}^{\\,\\,\\,\\,*}_{\\,\\,\\,2}x_{22}x^*_{22} + \\fte_{\\,\\,\\,1}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,1}x_{12}x^*_{12}.\n  \\end{array} \n\\end{equation}\n\n\nThe polarisation tensor can be related to the Stokes parameters in the standard way (see equation~\\ref{polarisation_tensor}), and using these relations, the Stokes parameters observed in frame $S'$, can be expressed in terms of the field components of ($\\bm{E}, \\bm{B}$), as calculated in frame $S$. In the same way, $(I',Q',U',V')$ in $S'$ can be related to $(I,Q,U,V)$ in frame $S$. Carrying out the algebra the results are as follows:\n\n\\noindent\n\\parbox{0.8\\columnwidth}{\n\\begin{eqnarray}\nI' & = &   \\frac{1}{2} \\left\\{ (x_{11}^2+x_{12}^2)(I+Q)+(x_{21}^2+x_{22}^2)(I-Q) \\right\\}  \\nonumber \\\\\nQ' & = &   \\frac{1}{2} \\left\\{ (x_{11}^2-x_{12}^2)(I+Q)-(x_{22}^2-x_{21}^2)(I-Q) \\right\\}  \\nonumber \\\\ \nU' & = &   \\hspace{1em}\\frac{}{} (x_{12}x_{11})(I+Q)+(x_{22}x_{21})(I-Q) \\nonumber \\\\ \nV' & = &   \\hspace{1.3em}(x_{22}x_{11}-x_{12}x_{21}) \\, V \\nonumber \n\\end{eqnarray}}\n\\hspace{0.3cm}\n\\parbox{0.05\\columnwidth}{$\\left.\n  \\begin{array}{c}   \n               \\\\ \\\\ \\\\  \\\\ \\\\ \\\\ \\\\                 \n  \\end{array}\n  \\right\\} $} \n\\hfill \n\\parbox{0.05\\columnwidth}{\\begin{eqnarray}\\label{test-label} \\end{eqnarray}}\n\nA significant source motion will alter the observed polarisation, the intensity and magnitude of the Stokes parameters will change, and this will be accompanied by a Doppler shift in the observed frequency. The appearance of a non-zero Stokes U in the observer frame is not in itself significant, as the relative magnitude of Stokes Q and U, depend on the orientation of your reference axes in the observer plane. Figure~\\ref{boosting_on_stokes} shows the source and observer polarisation spectrum for a source magnetic field of $1.9\\times10^6$ G, with a source velocity of $\\beta=0.87$.  \n\n\n\\section[4]{Description of the Computational Model}\n\\label{sec:4}\n\nThe above physical emission model is simulated within the pulsar's magnetospheric environment. The three dimensional magnetosphere is represented computationally in spherical polar coordinates, and it is assumed that emission arises from within the open volume of the magnetosphere, which is determined numerically. As such, emission is only simulated from those grid points lying in the open volume, where the grid itself is specified using coordinates $(r,\\theta,\\phi)$. The number of grid points on any given spherical surface is controlled by the parameters $N_\\theta$ and $N_\\phi$, (the number of $\\theta$ and $\\phi$ divisions). The parameter $N_r$ specifies the number of divisions in the radial direction, and the grid itself is devised in such a way that each grid point represents an equal volume of space, so that in essence, the discretisation of the magnetosphere is homogeneous with respect to volume. \n\nThe emission is calculated at all points in the open volume, which requires the determination of {\\it local} parameters. These include the magnetic field strength, $\\bm{B}$, the total number of particles at a specific point, $n_{GJ}$, the assumed pitch angle distribution, local co-rotational velocity and the range of particle gamma factors which contribute to observable emission at that particular point. All necessary dependencies are calculated at each grid point to create the local emission profile, which effectively determines how the polarised spectral emissivity varies across the extent of the local pitch angle distribution. Each profile is subject to the required frame transformations (described above), which determine the emissivity profile as seen by a stationary observer external to the rotating magnetosphere.\n\n\\begin{figure*}\n\\center\n{\\includegraphics[scale=0.4]{grid_mix_2.eps}} \n\\caption{Centre: The concentric spherical Cartesian grid imposed on the pulsar magnetosphere, showing points only in the open volume of an aligned rotator ($\\uv{\\Omega}\\cdot\\uv{\\mu}=1$). Right: At each individual location ($\\bm{R}$), we calculate the radiation emitted from the local particle distribution, taking the local physical conditions into account - e.g., $\\bm{B}, \\bm{v}_{co}$, the power law particle distribution, the specific pitch angle distribution etc. Left: Representation of how emission from a particle pitch angle distribution extends over a range of viewing angles ($\\chi$) and phase ($\\Phi$). Emission is recorded computationally into discrete bins of extent $\\Delta\\chi \\times \\Delta\\Phi$ (as described in section \\ref{sec:4}).}\n\\label{spherical_grid}\n\\end{figure*}\n\nThe pitch angle distribution at any open volume point, generates emission which extends over a finite solid angle. This solid angle of emission has a finite extent latitudinally and longitudinally with respect to an observer, which directly correspond to a range in observer viewing-angle, $\\chi$ and temporal phase, $\\Phi$, respectively. Discretising this observer space\\footnote{Representing $0\\deg \\le \\chi \\le 180\\deg$ and $0 \\le \\Phi \\le 1$ with step-sizes of $\\Delta \\chi$ and $\\Delta \\Phi$.} allows emission from any point to be recorded as a function of viewing angle and phase. For any given model pulsar (i.e., for a specific magnetic inclination, $\\alpha$), the code records the phase resolved polarimetry in the form of an array, denoted as $\\xi(R,\\theta,\\phi,\\Phi,\\alpha,\\chi)$ ($=\\xi(\\bm{R},\\Phi,\\alpha,\\chi)$). This means that for each grid point location, $\\bm{R}$, the intensity and phase resolved Stokes parameters for the range of viewing angles over which emission is seen, are recorded.\\footnote{The magnetosphere is of course rotating, so at any point in time, the correct phase of emission is determined taking light flight time, aberration and rotation into account.}\\\\\n\nThe indices in the $\\xi$ array are discrete and represents a record of the phase resolved Stokes parameters seen by observers from point $\\bm{R}$. Once all grid points have been sampled, the array $\\xi$ can be summed from all appropriate locations to form $\\xi(\\alpha, \\chi, \\Phi)$\\footnote{Represented in Figure~\\ref{flow.diagram} as the array $\\xi(\\alpha_A,\\chi_B,\\Phi)$.} - i.e., the phase resolved Stokes parameters observed from the entire magnetosphere for a pulsar inclination of $\\alpha$. \\\\\n\nA single run of this code will create files which contain $\\xi(\\alpha,\\chi,\\Phi)$ for a single, specific pulsar inclination $\\alpha$. The search algorithm approach attempts to restrict $\\alpha$ and $\\chi$ by simulating emission for the range of all {\\it possible} $\\alpha$ values\\footnote{By explicit assumption, there is no a priori restriction on $\\alpha$ (which is generally the case, as observations constrain neither $\\alpha$ nor $\\chi$).} ($0\\deg \\le \\alpha \\le 180\\deg$), and subsequently selecting those simulated lightcurves which best fit the observational data (which should correspond to a unique combination of $\\alpha$ and $\\chi$). To generate lightcurves for different possible inclinations of a given pulsar,  $\\alpha$ is varied from $0\\deg \\le \\alpha \\le 180\\deg$ in discrete steps $\\Delta \\alpha$. This generates a large pool of phase resolved model lightcurves, each dependant on a unique combination of $\\alpha$ and $\\chi$ - for example, with $\\Delta \\alpha = 10\\deg$ and $\\Delta\\chi=1\\deg$, one generates $180\\times18= 3240$ distinct lightcurves ($\\times 4$ when polarimetry is considered). \n\n\n\n\n\\section{Results}\n\nThe search algorithm approach was partly motivated on the premise that any simulated Stokes parameters $(\\xi)$, would be a variable function of the phase space $(\\alpha,\\chi,$ {\\small PAD}, {\\small PAD}$_{co}$). The simulated Stokes parameters vary as a smooth function of all phase space variables $\\xi(\\alpha,\\chi,$ {\\small PAD}, {\\small PAD}$_{co}$\\footnote{PAD refers to the pitch angle distribution which can either be isotropic or have a Gaussian structure; {\\small PAD}$_{co}$ defines the maximum  pitch angle.}  ), thereby allowing for the selection of a best fitting parameter set.\nThis being the case, one can restrict the pulsar geometry ($(\\alpha,\\chi)$ combination), which best fits observations and subsequently carry out the inverse mapping step to locate the associated source region within the pulsar magnetosphere.\\\\\n\nSimulated Stokes parameters are by definition dependent on all phase space variables $(\\alpha,\\chi,$ {\\small PAD},{\\small PAD}$_{co}$). As described in section \\ref{sec:4}, this phase space is sampled discretely, thereby generating a large amount of model lightcurves. We find that the simulated Stokes parameters ($\\xi$), vary smoothly as a function of this parameter space, showing systematic behaviour as we vary the observed energy band, viewing angle $\\chi$, pulsar inclination $\\alpha$, and particle pitch angle distributions. We discuss initially, the form of the simulated Stokes parameters, how these vary as a function of parameter space and how this variation can be used to select a best fitting parameter combination (the `search algorithm'). We subsequently describe our restrictions on pulsar geometry ($(\\alpha,\\chi)$ combination), and discuss the inverse mapping step which locates the associated source region within the pulsar magnetosphere.\\\\\n\nWe note here, that  our simulations resolve pulsar phase into 200 bins, and we subsequently quote the phase extent of a single pulsar cycle as going from 0 $\\rightarrow$ 1. As choice of a reference phase is arbitrary, we define phase 0 to be the phase at which the magnetic pole associated with the quoted ${\\alpha}$ value would be seen by an external observer. From symmetry, phase 0.5 is the phase at which the magnetic pole at ${\\alpha=180\\deg-\\alpha}$ would be seen. We refer to the magnetic pole at inclination $\\alpha$ as pole 1 and the magnetic pole at $180\\deg-\\alpha$ as pole 2.\n\n\\subsection{Form and variation of simulated Stokes parameters}\n\nThe search algorithm method, in sampling such a large parameter space, inherently creates a large amount of data. It is instructive initially to discuss the overall form of the simulated Stokes parameters and to describe how they vary as a function of the phase space parameters  $\\alpha, \\chi,$ {\\small PAD}, and {\\small PAD}$_{co}$. Some useful systematic trends are evident which will be discussed.\n\n\\subsubsection{{\\small PAD}$\\:$and {\\small PAD}$_{co}$}\n\nIt is found that similar trends exist for the variations of $\\xi$ between the different {\\small PAD } combinations (i.e., each of the {\\small PAD } functions based on isotropic, circular, linear and Gaussian profiles all have similar $\\xi(\\alpha,\\chi)$). We find that varying the {\\small PAD}$_{co}$  has a larger effect on the polarisation than varying the {\\small PAD}  itself. The general trend is for larger {\\small PAD}$_{co}$  values to generate broader peaks, but the peaks are located at the same phase locations so that the overall variation of $\\xi(\\alpha,\\chi)$ is similar for all {\\small PAD} s. Given this degenerate behaviour, we will focus our analysis on the isotropic pitch angle distribution, which should reflect the overall general trends in the variation of $\\xi(\\alpha,\\chi)$.\n\n\\subsubsection{Stokes I: Single and Double Peaks}\n\nWe have found that simulated lightcurves only have a single peak for any inclination $\\alpha \\lesssim 50\\deg$, but a double peak profile is seen to emerge for $\\alpha \\gtrsim 50\\deg$. The intensity of the secondary peak increases as $\\alpha \\rightarrow 90\\deg$, where the main:secondary peak ratio varies smoothly with viewing angle. This allows for an immediate restriction in terms of predicted inclination and viewing angle for single and double peaked pulsar profiles.\\\\\n\n\n\n \n\n\\begin{figure*}\n\\vspace{-10em}\n\\centering\n\\begin{minipage}{0.49\\linewidth}\n\\hspace{8em}\n\\includegraphics[viewport=110 300 500 690, width=1.0\\linewidth]{StokesI_70B.eps}\n\\end{minipage}\n\\begin{minipage}{0.49\\linewidth}\n\\hspace{8em}\n\\includegraphics[viewport=110 300 500 690, width=1.0\\linewidth]{StokesI_80B.eps}\n\\end{minipage}\n\\begin{minipage}{0.49\\linewidth}\n\\hspace{8em}\n\\includegraphics[viewport=110 300 500 690, width=1.0\\linewidth]{StokesI_90B.eps}\n\\end{minipage}\n\\vspace{14em}\n\\caption[Primary results 4]{Relative intensity (Stokes I) at all viewing angles (V. A.) for successive pulsar inclinations. Parameter sets are (clockwise from top left) $(\\alpha)= 70\\deg, 80\\deg, 90\\deg$ with ({\\small PAD}, {\\small PAD}$_{co}$)$=$(isotropic, $20\\deg$) in each case. The secondary peak increases its intensity and viewing angle extent, as $\\alpha \\rightarrow 90\\deg$. At $\\alpha=90\\deg$, the `secondary peak' is of equal intensity to the main peak. In terms of restricting parameter space, this evolution shows that the observed main:secondary peak ratio of $\\bm{ \\sim$ $\\bm 0.3}$ (for the Crab pulsar), can only be reproduced for a restricted range of viewing angles (for $\\bm{\\Delta \\chi \\sim 5\\deg}$ about a central $\\bm \\chi_o$) for $\\bm{ \\alpha \\gtrsim 60\\deg}$}.\n\\label{3d-peak2}\n\\end{figure*}\n\n\n\n\n\n\nUsing $\\xi$=I$(\\bm{R},\\alpha,\\chi,\\Phi)$, it is found that the main emission contribution to the secondary peak, comes from pole 2 if $ \\chi\\lesssim90\\deg$, and from pole 1 if $ \\chi\\gtrsim 90\\deg$, indicating that as $ \\alpha\\gtrsim 50\\deg$, the secondary peak is formed from the magnetic pole {\\it opposite} the pole contributing to the main peak. Figure~\\ref{10} shows some phase resolved plots of the I(pole 1):I(pole 2) ratios for an inclination of $\\alpha=80\\deg$, at different viewing angles, which illustrates the relative contributions of emission over each pole to the final integrated lightcurve.\\\\\n\n\\begin{figure}\n\\centering\n{\\includegraphics[scale=0.33,angle=-90,clip]{ratios_80_50.eps}}\n\\hfill\n{\\includegraphics[scale=0.33,angle=-90,clip]{ratios_80_130.eps}}\n\\caption[Primary results 6]{Intensity observed for $(\\alpha)=80\\deg$ showing relative relative flux contribution of emission from pole 1 and pole 2 for different viewing angles.  Here pole 1 refers to the magnetospheric regions above pole 1 and not just to the polar cap regions. Results illustrate that, depending on whether $\\chi\\gtlt90\\deg$, different poles will contribute to main and interpulse emission.}\n\\label{10}\n\\end{figure}\n\n\\subsection{General Observational Restrictions}\n\nCertain features of Crab pulsar emission are clear and distinct and any successful modelling of this object should be able to recreate at least some of these main features:\n\n\\newcounter{listno5}\n\\begin{list}{ [\\alph{listno5}] }\n  {\\usecounter{listno5}\n  {\\setlength{\\rightmargin}{0.1\\columnwidth}}\n  {\\setlength{\\leftmargin}{0.1\\columnwidth}}}\n\n\\item {\\bf Double peaks:} The Crab pulsar exhibits a double peaked structure at all wavelengths. Only simulated dipole inclinations with $\\alpha \\gtrsim 50\\deg$ are capable of producing double peak emission. This is our first restriction of phase space.\\\\\n\n\\item {\\bf Relative peak intensity and bridge emission:} The double peaked structure is accompanied by the requirements of explaining the relative intensity of the peaks. A secondary peak whose intensity and extent of visibility (range of $\\chi$ values over which it is visible) is a (monotonic) function of inclination $\\alpha$, appears within the simulations for $\\alpha \\gtrsim 50\\deg$. The ratio of main:secondary peak intensity of 1:0.3, is possible for a narrow range of $\\chi$ associated with any $\\alpha \\gtrsim 60\\deg$. This range of $\\chi$ extends no more than $\\sim 2.5\\deg$ about a central $\\chi=\\chi_o$. We find that $\\chi_o$ is a smooth function of pulsar inclination ($\\chi_o=\\chi_o(\\alpha)$), $\\chi_o$ decreasing steadily as $\\alpha$ increases.\\footnote{$\\chi_o \\gtrsim 80\\deg$ for $\\alpha = 60\\deg$, $\\chi_o \\sim 47\\deg$ for $\\alpha=80\\deg$, as can be seen in Figure~\\ref{3d-peak2}.} Our second restriction can be summarised as follows: only for $\\alpha \\gtrsim 60\\deg$ and for  viewing angles $\\Delta \\chi \\sim 2.5\\deg$ about $\\chi_o=\\chi_o(\\alpha)$, are simulated lightcurves able to reproduce the main:secondary peak ratio of $\\sim$ 1:0.3. Bridge emission is also a function of $\\alpha$ where the large bridge emission seen at lower inclinations ($\\alpha$), tends to disappear for the more orthogonal rotators. \\\\\n\n\n\\item {\\bf Peak phase separation:} The Crab pulsar's peak separation is $\\sim 0.4$ in phase. Results indicate that the simulated model is unable to produce a double peak structure separated by less than $0.4$ in phase, where in fact most viewing angles have peak separations of 0.5. In hindsight, this is a result of the inherent symmetry of the physical model, as it simulates emission from the open volume of both poles of a symmetric dipolar magnetic field structure. The influence of relativistic beaming can lead to asymmetries in pulse structures, so one would sensibly infer that a global emission model viewed at $\\alpha\\ne 90\\deg, \\chi \\ne 90\\deg$ should subsequently exhibit non-symmetric behaviour. It seems therefore, that the explanation for the symmetry in these model results, most likely lies in the averaging of emission over such a large volume. The large width of the peaks is also most likely due to the large volume of emission. It is hoped that the inclusion of more general physical constraints into the model, should lead to a removal of the symmetry.\\\\\n\n\\end{list}\n\n\\subsection{Inverse Mapping - A two pole emitter?}\n\n\nOur results given that the model results are capable of more successfully reproducing  lightcurve flux profiles than the associated polarisation \nprofiles, inverse mapping results are given for a selected ($\\alpha,\\chi$) combination, which compares most favourably to the observed \ndouble peak structure of the Crab pulsar and the observed QU relationship. A value of $(\\alpha,\\chi)=(70\\deg,45\\deg)$ is chosen \nfor ({\\small PAD}, {\\small PAD}$_{co}$)=(isotropic, $20\\deg$), these values were based on a best fit in the $\\chi^2$ sense between the simulated data and the \\cite{SJDP1988} \nobservations. In Figure \\ref{Phase_Plots} we show the intensity and degree of linear and circular polarisation as a function of phase. We note that the intensity distribution is wider than normal observed - see for example \\cite{2009MNRAS.397..103S} this can be explained by our initial approach of filling the open field region with radiating particles. \\cite{2009MNRAS.397..103S} also showed an increase in the degree of linear polarisation happening prior to the main pulse which we see although not at exactly the same phase relationship. From Figure \\ref{PA_Plots}  we see the expected swing in polarisation angle around the two peaks and QU plots morphologically similar to \\cite{2009MNRAS.397..103S} and \\cite{S1986}. Our preferred values of $\\alpha$ and $\\chi$  are broadly consistent with \\cite{2004ApJ...601..479N}, which provided a robust estimate of the pulsar inclination angle.\n\nThe inverse mapping element of the search algorithm allows us to decompose light-curves, with the option of selecting different phase resolved regions (on our current resolution phase is divided into 200 bins), which we can subsequently retrace back into the \\mag, isolating the originating locations of the component photons. This will localise the magnetospheric regions which contribute to emission at a particular phase, one of the main motivations for the search algorithm.\\\\\n\nWe try and present the results of the inverse mapping in an interpretable sense here in 2 and 3 dimensions, although the task is not an easy one. The selected $(\\alpha,\\chi,${\\small PAD},{\\small PAD}$_{co}$) combination chosen to illustrate the inverse mapping is $(70\\deg, 52\\deg,\\:$isotropic, $20\\deg$), which has a main peak at phase  $\\sim 0.1$ and an interpulse at phase $\\sim 0.6$. We choose specific phase regions ($\\Delta\\Phi=5$) centred on the arrival phase of the main peak and the inter-pulse, so as to obtain a view of the magnetospheric regions which contribute to `peak' intensity.We note however on the basis of Figure \\ref{10} that the fit is a smoothly varying function of $\\alpha,\\chi$,{\\small PAD} and {\\small PAD}$_{co}$. In Figure \\ref{POD1_Plot} we show the effect of changing the {\\small PAD}$_{co}$ showing intensity vs phase for {\\small PAD}$_{co}$ values of 1$\\deg$ and 5$\\deg$. \\\\ \n\n\\begin{figure*}\n\\centering\n{\\includegraphics[scale=0.65,angle=0,clip]{INT-both.eps}}\n\n\\hfill\n\n\\caption[Phase Plots]{Phase plots for 70$\\deg$ inclination and viewing angles 45$\\deg$ and 20$\\deg$. We show the total optical intensity, and degree of linear\/circular polarisation.  }\n\\label{Phase_Plots}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\n\n{\\includegraphics[scale=0.53,angle=0]{PA-both.eps}}\n\n\\caption[PA Plots]{Illustrative polarisation angle vs phase and QU plot  for 70$\\deg$ inclination and viewing angles 45$\\deg$ and 20$\\deg$. }\n\\label{PA_Plots}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{minipage}[a]{1.0\\linewidth}\n\\centering\n\\includegraphics[scale=0.6,angle=0]{POD_john.eps}\n\\caption[POD1 Plots]{Phase and intensity relationship for 70$\\deg$ inclination and viewing angle 45$\\deg$ for two ${\\small PAD}$ of 1 and 5 showing the ragged nature of the intensity for the lower pitch angles. }\n\\label{fig:figure1}\n\\end{minipage}\n\\hspace{0.5cm}\n\\label{POD1_Plot}\n\\end{figure*}\n\n\n\n\n\n\nCarrying out the inverse mapping step for $(\\alpha,\\chi)=(70\\deg,45\\deg)$, Figure~\\ref{main-peak-3d} presents a 3D colour coded intensity map of the originating location of emitted radiation seen by an observer at $\\chi=52\\deg$ relative to the model pulsar. Figure~\\ref{main-peak-3d} (left) shows the origin of emission which composes the main peak and Figure~\\ref{main-peak-3d} (right) shows the origin of emission composing the interpulse (as an aid to perspective, the circles define the light cylinder distance and the brownish field lines are the boundary of the open volume\/closed volume region within the pulsar magnetosphere). Immediate points of note, are that the main peak is primarily composed of emission from one pole (which we term pole 1), while the interpulse is composed of emission from the opposite pole (here defined as pole 2). Also, the most intense emission is located in tightly constrained volumes in the inner magnetosphere.\\\\ \n\n\n\n\\begin{figure*}\n\\includegraphics[scale=0.31,clip]{FinalImageB.eps}\n\\caption[3D representation of inverse mapped regions for main peak emission]{Three dimensional representation of the regions contributing to {\\bf main peak} (left) and {\\bf secondary peak} (right) emission for a model (Crab) pulsar with parameters ($\\alpha,\\chi$,{\\small PAD},{\\small PAD}$_{co}$)=($80\\deg,25\\deg,\\:$isotropic, $20\\deg$). The intensity scale shows the emission contributions at the regions mapped. For reference, the circles define the light cylinder boundary and the light brown lines are the open volume\/closed volume boundary. Emission is seen to be concentrated in the lower magnetosphere from a single pole.}\n\\label{main-peak-3d}\n\\end{figure*}\n\nIt is immediately apparent that these results show localised emission regions and clearly indicate a two pole emission model. Figure~\\ref{main-peak-3d} illustrates rather dramatically, that the first order assumption of a global truncated power law synchrotron emission model (even for a relatively wide pitch angle distribution of $20\\deg$) can produce a rather limited spatial extent of maximum emission within the pulsar magnetosphere, while matching certain features observed in the Crab pulsar profile. The main location of emission is well away from the standard outer gap location and closer to the polar cap centre, but a closer analysis (described subsequently) yields that the emission is at distances of $\\sim  0.2R_{LC}$ from the polar cap, further from the polar cap surface than polar cap models generally estimate. \\\\\n \n\nIn an independent analysis, we attempt to localise relative emission contribution as a function of magnetospheric location. We subdivide the magnetospheric volume into concentric spherical shells of width $\\Delta\\rho$ ($\\rho$ being the spherical radius), and also into coaxial cylindrical shells of thickness $\\Delta r$ ($r$ in cylindrical coordinates). We integrate emission from all points within these shells to obtain a radial profile (spherically and cylindrically respectively) of absolute contributions to the total emission. The spherical profile is presented in the left hand panels of Figures~\\ref{13} and \\ref{14} with the cylindrical contributions shown on the right. Figure~\\ref{13} shows emission contributions to the main emission peak at phase $\\sim 0.1$, Figure.~\\ref{14} shows emission contributions to the secondary peak at phase $\\sim 0.6$. The top panels in these figures show the total emission from over both magnetospheric poles, the middle panels show emission regions associated with pole 1\\footnote{Pole 1 corresponds to the magnetospheric region above the \nmagnetic pole at an inclination $\\alpha$} only and the bottom panels show emission from pole 2\\footnote{Pole 2 refers to emission\nassociated with the pole at inclination $180\\deg + \\alpha$.} only.\\\\\n\nFrom the previous figures, certain trends are apparent. Pole 1 emission dominates for the main pulse and pole 2 emission for the secondary, each pole having distinct emissivity trends, both radially (cylindrically) and spherically. The poles dominating emission tend to have their maximum contributions located at very small spherical and cylindrical ($\\sim 0.2 R_{LC}$) radii, whereas the poles which contribute less to the total emission are located further from the star. The high peak in the emissivity, may be due to the influence of higher local charge densities lower down in the magnetosphere. Generally, the collective interpretation is that the main contribution to emission is very localised closer to the neutron star surface\\footnote{The poles contributing less to the emission of a given peak also seem to have their emission localised, but at a greater distance from the star.} than to the light cylinder. This tends to agree with models attempting to explain optical emission through a localised effect such as the slot gap.  Indeed our results are consistent with the emission altitude predicted by \\cite{2010MNRAS.405..509D}. Furthermore we note that the slot-gap models primarily localise emission in the transverse (co-latitude) direction whereas we also restrict emission in altitude.\n\n   \n\\subsection{Conclusion}\n\nOptical polarisation studies can be used to determine the local geometry of the emission region. In particular from this work we have\n\\begin{itemize}\n\\item A simple synchrotron model for the emission gives reasonable agreement with observations.\n\\item A prediction that the emission is low in the magnetosphere at an altitude in the region 30-40 stellar radii, that is away from both the polar cap and likely outer gap regions. \n\\item The linear polarisation peaks on the rising edge of the main pulse, consistent with the findings of \\cite{SJDP1988,2009MNRAS.397..103S}.\n\\item The radiation for the most part is circularly polarised with an interplay between  linear and circular polarisation on the rising edge of the main pulse.  On this basis observations of pulsed circular polarisation from optical pulsars would provide a significant geometrical restrict on pulsar parameters although we accept more detail work is needed of more realistic emission zones. \n\\item Pulse widths which are significantly greater than those observed although this probably stems from our requirement to fill to open field region with an emitting plasma.\n\\item Due the inherent symmetry of this model we do not see any significant bridge emission at the preferred orientation and viewing angle. \n\n\n\\end{itemize}\n\nOur future work will entail restricting emissions to localities around the last open field lines thereby removing the inherent symmetry existing within the model. In this way we hope to able to make firmer predictions in relation to the optical emission specifically and more generally comment on correlations between the optical and radio emission \\cite{2009MNRAS.397..103S}.  As well as the emission location and pulsar orientation our approach can also restrict the {\\small PAD} and {\\small PAD}$_{co}$. We also intend to make more detailed comparison between these predictions and measurements of the pulsed circular polarisation. To-date there have been no measurements of optical circular polarisation from any pulsar.  A new instrument, the Galway Astronomical Stokes Polarimeter (GASP), \\cite{2010EPJWC...505003K} has this capability and observations are planned to observe the Crab pulsar in late 2011.\n\n\n\n\\section*{Acknowledgments}\nThe authors would like to thank the Higher Education Authority Programme for Research in Third Level Institutions (PRTLI) Cosmogrid project and National University of Ireland's Millenium Fund for financial support for POC for the duration of this work and the PRTLI 4 E-Inis project for financial support for JMcD for the duration of this work. Finally we would also like to thanks the anonymous referee whose comments made significant improvements to an earlier version of this paper.\n\n\n\\pagebreak\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe fix a nonnegative real number $T$ as well as two positive integers $k$ and $d$, and let $\\R^+:=[0,+\\infty)$. Let $\\mathbbm{1}_{A}$ represent the indicator function of a set $A$, and $\\langle x,y\\rangle$ the inner product of $x,y\\in\\R^k$. The Euclidean norms of a vector $y\\in\\R^k$ and a matrix $z\\in \\R^{k\\times d}$ are defined by $|y|$ and $|z|$, respectively.\n\nAssume that $(\\Omega,\\F,\\mathbbm{P})$ is a completed probability space carrying a standard $d$-dimensional Brownian motion $(B_t)_{t\\geq 0}$, and that $(\\F_t)_{t\\geq 0}$ is the natural $\\sigma$-algebra filtration generated by $(B_t)_{t\\geq 0}$ and $\\F=\\F_T$. For each $p>0$, denote by $\\Lp$ the set of all $\\R^k$-valued and $\\F_T$-measurable random vectors $\\xi$ such that $\\E[|\\xi|^p]<+\\infty$, by ${\\s}^p(0,T;\\R^k)$ (or $\\s^p$ simply) the set of $\\R^k$-valued, $(\\F_t)$-adapted and continuous processes $(Y_t)_{t\\in\\T}$ such that\n$$\\|Y\\|_{{\\s}^p}:=\\left( \\E\\left[\\sup_{t\\in\\T} |Y_t|^p\\right] \\right)^{1\\wedge 1\/p}<+\\infty,$$\nand by ${\\rm M}^p(0,T;\\R^{k\\times d})$ (or ${\\rm M}^p$ simply) the set of $(\\F_t)$-progressively measurable ${\\R}^{k\\times d}$-valued processes $(Z_t)_{ t\\in\\T}$ such that\n$$\\|Z\\|_{{\\rm M}^p}:=\\left\\{ \\E\\left[\\left(\\int_0^T |Z_t|^2\\ {\\rm d}t\\right)^{p\/2}\\right] \\right\\}^{1\\wedge 1\/p}<+\\infty.\n$$\nIt is well known that for each $p\\geq 1$, $\\s^p$ and ${\\rm M}^p$ are both Banach spaces respectively endowed with the norms $\\|\\cdot\\|_{{\\s}^p}$ and $\\|\\cdot\\|_{{\\rm M}^p}$. And, for each $p\\in (0,1)$, $\\s^p$ and ${\\rm M}^p$ are both complete metric spaces with the resulting distances $(Y,Y')\\mapsto \\|Y-Y'\\|_{{\\s}^p}$ and $(Z,Z')\\mapsto \\|Z-Z'\\|_{{\\rm M}^p}$ respectively.\n\nWe recall that a process $(Y_t)_{t\\in\\T}$ belongs to the class (D) if the family of variables $\\{|Y_\\tau|:\\tau\\ {\\rm is\\ an}$ $(\\F_t){\\rm - stopping\\ time\\ bounded\\ by}\\ n\\}$\nis uniformly integrable.\n\nIn this paper, we are interested in solving the following multidimensional backward stochastic differential equation (BSDE for short):\n\\begin{equation}\ny_t=\\xi+\\int_t^Tg(s,y_s,z_s){\\rm d}s-\\int_t^Tz_s {\\rm d}B_s,\\ \\   t\\in\\T,\n\\end{equation}\nwhere $\\xi\\in\\LT$ is called the terminal condition, $T$ is called the time horizon, and the random function $$g(\\omega,t,y,z):\\Omega\\tim \\T\\tim {\\R}^{k }\\tim\n{\\R}^{k\\times d}\\longmapsto {\\R}^k$$\nis $(\\F_t)$-progressively measurable for each $(y,z)$, called the generator of BSDE (1). Furthermore, the triple $(\\xi,T,g)$ is usually called the parameters of BSDE (1).\\vspace{0.1cm}\n\nThroughout this paper, we use the following definitions on solutions of (1).\\vspace{0.1cm}\n\n{\\bf Definition 1}\\ \\ A solution of BSDE (1) is a pair of $(\\F_t)$-progressively measurable processes $(y_t,z_t)_{t\\in\\T}$ with values in ${\\R}^k\\times {\\R}^{k\\times d}$ such that $\\ps$, $\\int_0^T|z_t|^2\\ {\\rm d}t<+\\infty$, $\\int_0^T|g(t,y_t,z_t)|\\ {\\rm d}t<+\\infty$, and (1) holds true for each $t\\in\\T$.\\vspace{0.2cm}\n\n{\\bf Definition 2}\\ \\ Assume that $(y_t,z_t)_{t\\in\\T}$ is a solution of (1). If $(y_t,z_t)_{t\\in\\T}\\in {\\s}^p(0,T;\\R^{k})\\times {\\rm M}^p(0,T;\\R^{k\\times d})$ for some $p>1$, then it is called an $L^p$ solution of BSDE (1); if $(y_t)_{t\\in\\T}$ belongs to the class (D) and\n$(y_t,z_t)_{t\\in\\T}\\in {\\s}^\\beta(0,T;\\R^{k})\\times {\\rm M}^\\beta(0,T;\\R^{k\\times d})$ for each $\\beta\\in (0,1)$, then it is called an $L^1$ solution of BSDE (1).\n\nIt is well known that nonlinear BSDEs were initially introduced in 1990 by \\citet{Par90}. They put forward and proved an existence and uniqueness result for $L^2$ solution of multidimensional BSDEs under the Lipschitz assumption of $g$ as well as the square integrability assumption of $\\xi$ and $g(t,0,0)$. From then on, the BSDE theory has attracted more and more interests, and due to the closely connections with many questions, it has gradually become a very powerful tool in many fields including stochastic control, financial mathematics, nonlinear mathematical expectation and partial differential equations, see \\cite{Bah10,Buc00,Chen00,Delb10,El97,Hu11,Hu08,Jia10,Kob00,\nMor09,Par99,Peng97,Tang98,Xing12} and so on.\n\nThere is no doubt that the existence and uniqueness of the solution is one of the most fundamental and kernel problems in the study on the theory and application of BSDEs. From the beginning, many researchers have attempted to improve the result of the $L^2$ solution of \\cite{Par90} by weakening the Lipschitz hypothesis on $g$, see, for example, \\cite{Bri07,Chen00,El97,Fan10,\nFan13,FJD13,Ham03,Jia08,Jia10,Lep97,Mao95,Xu15} for a survey.\nAt the same time, the existence and uniqueness of the $L^p\\ (p>1)$ solution for BSDEs has been extensively investigated by \\cite{Bri03,El97,Fan15,FJ14}, etc. Starting around 1998, the existence and uniqueness of the bounded solution and the solution whose exponential moments of certain order exist have also been becoming one of emphasis in the study on BSDE theory, one can see \\cite{Bri06,Bri08,Delb11,Hu15,Kob00,Lep98,Mor09,Rich12} for this topic, where the generator $g$ may have a quadratic or superquadratic growth in $z$.\n\nOn the other hand, in 1997, \\citet{Peng97} introduced the notion of $g$-martingales by solutions of BSDEs, which can be viewed as some kind of nonlinear martingales. Since the classical theory of martingales is carried in the integrable space, the question of solving a BSDE with only integrable parameters comes up naturally. In this spirit, some recent works including \\cite{Bri02,Bri03,Bri06,Fan12,FL10,Peng97,Xiao12,Xing12} investigated the existence and uniqueness of the $L^1$ solution of BSDEs. In particular, we would like to mention that \\citet{Bri03} established a general existence and uniqueness result of $L^1$ solution for a multidimensional BSDE with generator $g$ satisfying a monotonicity condition (see (H1) with $\\rho(x)=\\mu x$ in Section 3) as well as a general growth condition in $y$ (see (H2) in Section 3), and a Lipschitz condition together with a sublinear growth condition in $z$ (see (H3) in Section 3). Here, we also mention that multidimensional BSDEs are more difficult to handle than the one-dimensional case since for multidimensional BSDEs we usually can not establish or employ the comparison theorem of solutions. And, it is well known that the $L^1$ solution is more difficult to treat than the $L^p\\ (p>1)$ solution.\n\nThe present paper focus on the $L^1$ solution of multidimensional BSDEs. First of all, we will extend the existence and uniqueness result of the $L^1$ solution established in \\cite{Bri03} by weakening the monotonicity condition of $g$ in $y$ to a one-sided Osgood condition (see (H1) in Section 3). Under a Osgood condition of $g$ in $y$ and a Lipschitz condition of $g$ in $z$, \\citet{FJD13} first proved the existence and uniqueness of $L^2$ solution for multidimensional BSDEs. Recently, \\citet{Fan15} further extended this result and established the existence and uniqueness of $L^p\\ (p>1)$ solution for a multidimensional BSDE with generator $g$ satisfying a $p$-order weak monotonicity condition (see (H1a)$_p$ in Section 2) as well as a general growth condition in $y$, and a Lipschitz condition in $z$. We point out that in the case of $p=1$, the $p$-order weak monotonicity condition used in \\cite{Fan15} becomes the one-sided Osgood condition used in this paper. From this point of view, it is very natural to investigate the $L^1$ solution of multidimensional BSDEs under the one-sided Osgood condition of $g$ in $y$. However, when we use this condition instead of the usual monotonicity condition, some essential difficulty arise especially in the proof of existence of the $L^1$ solution. By virtue of Gronwall's inequality, Bihari's inequality and the relationship between convergence in $L^1$ and in probability together with two updated apriori estimates established in \\citet{Fan15}, we first consider the case when the generator $g$ is independent of $z$ (see Proposition 5 in Section 3). Then, making use of two estimates established in \\citet{Xu15} and \\citet{FJ14} together with Bihari's inequality, by a delicate argument involved in a Piciard's iterative procedure and a technique dividing the time interval $\\T$ we prove the existence of the $L^1$ solution for the general case (see Theorem 2 in Section 3). Here, we mention that it is interesting that in the case of the $\\alpha$ defined in (H3) values in $[1\/2,1)$, the one-sided Osgood condition need to be replaced with a $p$-order ($p>1$) one-sided Mao's condition (see (H1b)$_p$ in Section 2).\n\nThe second objective of this paper is to put forward and prove a stability theorem for the $L^1$ solutions of multidimensional BSDEs with generators of one-sided Osgood type. To the best of our knowledge, this is the first time for the $L^1$ solution of multidimensional BSDEs. It is not very hard to obtain a stability result of $L^p\\ (p>1)$ solutions for multidimensional BSDEs since by classical techniques one can establish and employ apriori estimates on the $L^p$ solution when $p>1$ (see, for example, Theorem 2 in \\citet{Fan15} for more details). However, it is well known that when $p=1$ the apriori estimates with respect to the first component of the $L^1$ solution are not valid any longer especially when the generator $g$ depends on $z$, which brings intrinsic difficulty when one tries to establish the stability of $L^1$ solutions. This may be the reason that by far there is still no reported work on the stability of $L^1$ solutions for multidimensional BSDEs even when $g$ only satisfies the monotonicity condition or the uniformly Lipschitz condition in $y$ other than the one-sided Osgood condition. In this paper, we will fill up the gap. More specifically, enlightened by the proof of the existence of the $L^1$ solution in this paper, we will first introduce some auxiliary BSDEs by virtue of a Picard's iterative procedure as a bridge and then use a very delicate argument to establish a stability theorem of the $L^1$ solutions for multidimensional BSDEs with generators of one-sided Osgood type (see Theorem 4 in Section 5), where expect for Gronwall's inequality, Bihari's inequality, the relationship between convergence in $L^1$ and in probability, and the technique dividing the time interval, the induction technique and the sharp apriori estimates established in \\cite{Fan15}, \\cite{Xu15} and \\cite{FJ14} all play important roles.\n\nIn addition, in this paper we also investigate, for the first time, the existence and uniqueness together with the stability of the solutions in the space $\\s^1\\times {\\rm M}^1$ for multidimensional BSDEs (see Theorem 3 in Section 4 and Theorem 5 in Section 5).\n\nThe remainder of this paper is organized as follows. In Section 2 we gather several updated apriori estimates with respect to the solutions of multidimensional BSDEs and two technical lemmas. In section 3 we state and prove the existence and uniqueness result of $L^1$ solutions for the multidimensional BSDEs, and in Section 4 we are interested in solving the multidimensional BSDEs in $\\s^1\\times {\\rm M}^1$ and provide two examples to illustrate our theoretical results. Finally, in Section 5 we put forward and prove a stability theorem of $L^1$ solutions as well as solutions in $\\s^1\\times {\\rm M}^1$ for the multidimensional BSDEs.\n\n\n\\section{Preliminaries}\n\nIn this section, we first introduce several sharp apriori estimates with respect to solutions of multidimensional BSDEs, which will play very important roles in the proof of our main results. For this, let us introduce the following assumptions with respect to the generator $g$:\\vspace{0.3cm}\n\n{\\bf (A1)}\\ \\ $\\as,\\ \\RE\\ (y,z)\\in \\R^k\\times\\R^{k\\times d},\\ \\ \\left\\langle y,\ng(\\omega,t,y,z)\\right\\rangle\\leq \\mu |y|^2+\\nu |y||z|+|y|f_t+\\varphi_t,\\ \\ $\\vspace{0.3cm}\\\\\nwhere $\\mu$ and $\\nu$ are two positive constants, $f_t$ and $\\varphi_t$ are two $(\\F_t)$-progressively measurable and nonnegative processes satisfying\n$$\n\\E\\left[\\left(\\int_0^T f_t\\ {\\rm d}t\\right)^p\\right]<+\\infty\\ \\ \\ {\\rm and}\\ \\ \\ \\E\\left[\\left(\\int_0^T \\varphi_t\\ {\\rm d}t\\right)^{p\/2}\\right]<+\\infty.\\vspace{0.2cm}\n$$\n\n{\\bf (A2)}\\ \\ $\\as,\\ \\RE\\ (y,z)\\in \\R^k\\times\\R^{k\\times d}$,\\vspace{0.25cm}\\\\\n\\hspace*{3cm}$|y|^{p-1}\\left\\langle {y\\over |y|}\\mathbbm{1}_{|y|\\neq 0},\ng(\\omega,t,y,z)\\right\\rangle\\leq \\psi(|y|^p)+\\nu |y|^{p-1}|z|+|y|^{p-1}f_t,$\\vspace{0.35cm}\\\\\nwhere $\\nu>0$ is a constant, $f_t$ is an $(\\F_t)$-progressively measurable and nonnegative process satisfying\n$$\\E\\left[\\left(\\int_0^T f_t\\ {\\rm d}t\\right)^p\\right]<+\\infty,$$\nand $\\psi(\\cdot):\\R^+\\mapsto \\R^+$ is a nondecreasing and concave function with $\\psi(0)=0$.\\vspace{0.3cm}\n\n{\\bf (A3)}\\ \\ $\\as,\\RE\\ (y,z)\\in \\R^k\\times\\R^{k\\times d},\\ \\ \\left\\langle {y\\over |y|}\\mathbbm{1}_{|y|\\neq 0}, g(\\omega,t,y,z)\\right\\rangle\\leq\n\\phi^{{1\\over p}}(|y|^p)+\\nu|z|+f_t,\\ \\ $\\vspace{0.3cm}\\\\\nwhere $\\nu>0$ is a constant, $f_t$ is an $(\\F_t)$-progressively measurable and nonnegative process satisfying\n$$\\E\\left[\\left(\\int_0^T f_t\\ {\\rm d}t\\right)^p\\right]<+\\infty,$$\nand $\\phi(\\cdot):\\R^+\\mapsto \\R^+$ is a nondecreasing and concave function with $\\phi(0)=0$.\\vspace{0.2cm}\n\nThe above assumptions (A2) and (A3) are respectively related to  following assumptions (H1a)$_p$ and (H1b)$_p$, which are put forward and used in \\citet{Fan15} at the first time. Assumptions (H1a)$_p$ and (H1b)$_p$ will also be employed in this paper.\\vspace{0.2cm}\n\n{\\bf (H1a)$_p$} $g$ satisfies a $p$-order weak monotonicity condition in $y$, i.e., there exists a nondecreasing and concave function $\\kappa(\\cdot):\\R^+\\mapsto \\R^+$ with $\\kappa(0)=0$, $\\kappa(u)>0$ for $u>0$ and $\\int_{0^+} {{\\rm d}u\\over \\kappa(u)}=+\\infty$ such that $\\as$, $\\RE\\ y_1,y_2\\in \\R^k,z\\in\\R^{k\\times d}$,\n$$\n|y_1-y_2|^{p-1}\\langle {y_1-y_2\\over |y_1-y_2|}\\mathbbm{1}_{|y_1-y_2|\\neq 0},g(\\omega,t,y_1,z)-g(\\omega,t,y_2,z)\\rangle\\leq \\kappa(|y_1-y_2|^p),\n$$\nwhere and hereafter, $\\int_{0^+} {{\\rm d}u\\over \\kappa(u)}:=\\lim\\limits_{\\epsilon \\rightarrow 0}\\int_{0}^{\\epsilon} {{\\rm d}u\\over \\kappa(u)};\\vspace{0.2cm}$\n\n{\\bf (H1b)$_p$} $g$ satisfies a $p$-order one-sided Mao's condition in $y$, i.e., there exists a nondecreasing and concave function $\\varrho(\\cdot):\\R^+\\mapsto \\R^+$ with $\\varrho(0)=0$, $\\varrho(u)>0$ for $u>0$ and $\\int_{0^+} {{\\rm d}u\\over \\varrho(u)}=+\\infty$ such that $\\as$, $\\RE\\ y_1,y_2\\in \\R^k,z\\in\\R^{k\\times d}$,\n$$\n\\langle {y_1-y_2\\over |y_1-y_2|}\\mathbbm{1}_{|y_1-y_2|\\neq 0},\ng(\\omega,t,y_1,z)-g(\\omega,t,y_2,z)\\rangle\\leq \\varrho^{{1\\over p}}(|y_1-y_2|^p).\\vspace{0.2cm}\n$$\n\nThe following Propositions 1-2 are respectively Propositions 2-3 in \\citet{Fan15}, and the following Proposition 3 comes from Proposition 1 in \\citet{Xu15}.\\vspace{0.1cm}\n\n{\\bf Proposition 1} Let $p>0$ and (A1) hold. Suppose that $(y_t,z_t)_{t\\in\\T}$ is a solution of BSDE (1) such that $y_\\cdot\\in {\\s}^p(0,T;\\R^{k})$. Then $z_\\cdot$ belongs to ${\\rm M}^p(0,T;\\R^{k\\times d})$, and $\\ps$, for each $0\\leq u\\leq t\\leq T$,\n$$\n\\begin{array}{lll}\n\\Dis \\E\\left[\\left.\\left(\\int_t^T |z_s|^2\\ {\\rm\nd}s\\right)^{p\/2}\\right|\\F_u\\right]&\\leq & \\Dis\nC_{\\mu,\\nu,p,T}\\E\\left[\\left.\\sup\\limits_{s\\in\n[t,T]}|y_s|^p\\right|\\F_u\\right]+C_p\\E\\left[\\left.\n\\left(\\int_t^T f_s\\ {\\rm d}s\\right)^p\\right|\\F_u\\right]\\vspace{0.1cm}\\\\\n&& \\Dis+C_p\\E\\left[\\left.\\left(\\int_t^T \\varphi_s\\ {\\rm d}s\\right)^{p\/2}\\right|\\F_u\\right],\n\\end{array}\n$$\nwhere $C_{\\mu,\\nu,p,T}>0$ is a constant depending on $(\\mu,\\nu,p,T)$, and $C_p>0$ is a constant depending only on $p$.\\vspace{0.3cm}\n\n{\\bf Proposition 2} Let $p>1$ and (A2) hold. Suppose that $(y_t,z_t)_{t\\in\\T}$ is an $L^p$ solution of BSDE (1). Then, there exists a constant $C_{\\nu,p}>0$ depending only on $\\nu, p$ such that $\\ps$, for each $0\\leq u\\leq t\\leq T$,\\vspace{0.2cm}\n$$\n\\E\\left[\\left.\\sup\\limits_{s\\in\n[t,T]}|y_s|^{p}\\right|\\F_u\\right]\\leq\ne^{C_{\\nu,p}(T-t)}\\left\\{\\E[\\left.|\\xi|^p\\right|\\F_u]+\\int_t^T\n\\psi(\\E[\\left.|y_s|^p\\right|\\F_u]) \\ {\\rm\nd}s+\\E\\left[\\left.\\left(\\int_t^T f_s\\ {\\rm\nd}s\\right)^{p}\\right|\\F_u\\right]\\right\\}.\\vspace{0.3cm}\n$$\n\n{\\bf Proposition 3} Let $g$ satisfy (A2) with $p=2$.  Suppose that $(y_t,z_t)_{t\\in\\T}$ is an $L^2$ solution of BSDE (1). Then, there exists a constant $C_{\\nu}>0$ depending only on $\\nu$ such that $\\ps$, for each $0\\leq u\\leq t\\leq T$,\\vspace{0.1cm}\n$$\n\\begin{array}{lll}\n&&\\Dis \\E\\left[\\left.\\sup\\limits_{r\\in\n[t,T]}|y_r|^2\\right|\\F_u\\right]+\\E\\left[\\left.\\int_t^T\n|z_s|^2\\ {\\rm d}s\\right|\\F_u\\right]\\vspace{0.1cm}\\\\\n&\\leq & \\Dis\ne^{C_{\\nu}(T-t)}\\left\\{\\E\\left[\\left.|\\xi|^2\\right|\\F_u\\right]+\n\\int_t^T\n\\psi\\left(\\E\\left[\\left.|y_s|^2\\right|\\F_u\\right]\\right)\\ {\\rm d}s+\\E\\left[\\left.\\left(\\int_t^T f_s\\ {\\rm\nd}s\\right)^2\\right|\\F_u\\right]\\right\\}.\n\\end{array}\\vspace{0.2cm}\n$$\n\nIn the same way as that in Lemmas 2-3 of \\citet{FJ14}, we can prove the following proposition. The proof is omitted here.\\vspace{0.1cm}\n\n{\\bf Propositions 4}\\ Let $p>1$ and (A3) hold. Suppose that  $(y_t,z_t)_{t\\in\\T}$ is a solution of BSDE (1) such that $y_\\cdot\\in {\\s}^p(0,T;\\R^{k})$. Then, $z_\\cdot$ belongs to ${\\rm M}^p(0,T;\\R^{k\\times d})$ and there exists a positive constant $C_{\\nu,p}$ depending on $\\nu$ and $p$ such that $\\ps$, for each $0\\leq u\\leq t\\leq T$,\\vspace{0.1cm}\n$$\n\\begin{array}{lll}\n&&\\Dis \\E\\left[\\left.\\sup\\limits_{s\\in\n[t,T]}|y_s|^{p}\\right|\\F_u\\right]+\\E\\left[\\left.\n\\left(\\int_t^T|z_s|^2\\ {\\rm d}s\\right)^{p\/2}\\right|\\F_u\\right]\\vspace{0.1cm}\\\\\n&\\leq & \\Dis e^{C_{\\nu,p}(T-t)}\\left\\{\\E\\left[\\left.|\\xi|^p\\right|\n\\F_u\\right]+\\int_t^T\n\\phi\\left(\\E\\left[\\left.|y_s|^p\\right|\\F_u\\right]\\right) \\ {\\rm d}s+\\E\\left[\\left.\\left(\\int_t^T f_s\\ {\\rm\nd}s\\right)^p\\right|\\F_u\\right]\\right\\}.\n\\end{array}\\vspace{0.3cm}\n$$\n\nNow, let us introduce two technical lemmas, which will be used later.  Firstly, the following Lemma 1 gives a sequence of upper bounds for linear growth functions, which comes from \\citet{Fan10}.\\vspace{0.2cm}\n\n{\\bf Lemma 1}\\ Suppose that $\\bar\\kappa(\\cdot):\\R^+\\mapsto \\R^+$ increases at most linearly, i.e., there exists a constant $A>0$ such that $$\\bar\\kappa(x)\\leq A(x+1),\\ \\ \\RE\\ x\\in \\R^+.$$\nThen for each $m\\geq 1$, we have\n$$\n\\bar\\kappa(x)\\leq (m+2A)x+\\bar\\kappa\\left({2A\\over m+2A}\\right),\\ \\ \\RE\\ x\\in \\R^+.\\vspace{0.2cm}\n$$\n\nThe following Lemma 2 can be regarded as a backward version of classical Bihari's inequality, which can be proved by classical methods. The proof is omitted.\\vspace{0.2cm}\n\n{\\bf Lemma 2}\\ (Bihari's inequality)\\ Let the nonnegative function $u(\\cdot):[0,T]\\mapsto \\R^+$ satisfy\n$$\nu(t)\\leq u_0 +\\int_t^T \\bar\\psi(u(s))\\ {\\rm d}s,\\ \\ t\\in \\T,\n$$\nwhere $u_0$ is a positive real number, $\\bar\\psi(\\cdot):\\R^+\\mapsto \\R^+$ is a continuous and nondecreasing function, $\\bar\\psi(0)=0$, $\\bar\\psi(u)>0$ for $u>0$ and $\\int_{0^+} {1\\over \\bar\\psi(u)}\\ {\\rm d}u=+\\infty$. Then, for each $t\\in \\T$, we have\n$$\nu(t)\\leq \\Psi^{-1}(\\Psi(u_0)+T-t),\n$$\nwhere\n$$\n\\Psi(x):=\\int_1^x {1\\over \\bar\\psi(u)}\\ {\\rm d}u, \\ \\ x>0\\vspace{0.2cm}\n$$\nis a strictly increasing function valued in $\\R$, and $\\Psi^{-1}$ is the inverse function of $\\Psi$. In particular, if $u_0=0$, then $u(t)=0$ for each $t\\in \\T$.\\vspace{0.2cm}\n\nTo the end of this section, we would like to especially mention that even though Propositions 1-4 mentioned above appear similar, there are some distinguish differences among both their conditions and conclusions. They will also play different roles in the proof of our main results.\n\n\n\n\\section{Existence and uniqueness of the $L^1$ solution}\n\nWe first introduce the following assumptions\non the generator $g$ used in \\citet{Fan15}, \\citet{Fan13} and \\citet{Bri03}:\\vspace{0.2cm}\n\n{\\bf (H1)} $g$ satisfies a one-sided Osgood condition in $y$, i.e., there exists a nondecreasing and concave function $\\rho(\\cdot):\\R^+\\mapsto \\R^+$ with $\\rho(0)=0$, $\\rho(u)>0$ for $u>0$ and $\\int_{0^+} {{\\rm d}u\\over \\rho(u)}=+\\infty$ such that $\\as$,\n$$\n\\RE\\ y_1,y_2\\in \\R^k,z\\in\\R^{k\\times d},\\ \\ \\left\\langle {y_1-y_2\\over |y_1-y_2|}\\mathbbm{1}_{|y_1-y_2|\\neq 0},\ng(\\omega,t,y_1,z)-g(\\omega,t,y_2,z)\\right\\rangle\\leq \\rho(|y_1-y_2|).\\vspace{0.2cm}\n$$\n\n{\\bf (H2)} $g$ has a general growth with respect to $y$, i.e, \\vspace{0.2cm}\n$$\\RE\\ r>0,\\ \\ \\E\\left[\\int_0^T \\bar\\phi_r(t)\\ {\\rm d}t\\right]<+\\infty\\ \\ {\\rm with}\\ \\ \\bar\\phi_r(t):=\\sup\\limits_{|y|\\leq r}\n|g(\\omega,t,y,0)|;$$\nFurthermore, $\\as$, $\\RE\\ z\\in {\\R^{k\\times d}},\\ \\ \\ y\\longmapsto g(\\omega,t,y,z)$ is continuous.\\vspace{0.2cm}\n\n{\\bf (H3)}\\ $g$ is Lipschitz continuous in $z$, uniformly with respect to $(\\omega,t,y)$, i.e., there exists a constant $\\lam\\geq 0$ such that $\\as$,\n$$\\RE\\ y\\in \\R^k,z_1,z_2\\in\\R^{k\\times d},\\ \\ |g(\\omega,t,y,z_1)-g(\\omega,t,y,z_2)|\\leq \\lam |z_1-z_2|;$$\nFurthermore, $g$ has a sublinear growth in $z$, i.e., there exist two constants $\\gamma>0$ and $\\alpha\\in (0,1)$ as well as an $(\\F_t)$-progressively measurable and nonnegative process $(g_t)_{t\\in\\T}$ satisfying $\\E\\left[\\int_0^Tg_t\\ {\\rm d}t\\right]<+\\infty$ such that $\\as$, \\vspace{0.05cm}\n$$\n\\RE\\ y\\in \\R^k,z\\in\\R^{k\\times d},\\ \\ |g(\\omega,t,y,z)-g(\\omega,t,y,0)|\\leq \\gamma(g_t(\\omega)+|y|+|z|)^\\alpha.\\vspace{0.2cm}\n$$\n\n{\\bf Remark 1} For later use, it follows from Proposition 1 in \\citet{Fan15} that for each $p>1$,\n$${\\rm (H1b)}_p \\Longrightarrow {\\rm (H1)}\\Longrightarrow {\\rm (H1a)}_p,$$\nand when $p=1$, they are same. In addition, the functions $\\rho(\\cdot)$, $\\kappa(\\cdot)$ and $\\varrho(\\cdot)$ in (H1), (H1a) and (H1b) all increase at most linearly since they are all nondecreasing and concave function valued $0$ at $0$. Here and hereafter we will always denote by $A$ the linear-growth constants of them, i.e.,\n$$\\rho(x)\\leq A(x+1),\\ \\ \\kappa(x)\\leq A(x+1),\\ \\ \\varrho(x)\\leq A(x+1),\\ \\ \\RE\\ x\\in\\R^+.$$\nFinally, by Proposition 1 in \\citet{Fan15} we also point out that the concavity condition of $\\rho(\\cdot)$ and $\\varrho(\\cdot)$ defined respectively in assumptions (H1) and (H1b)${}_p$ can be replaced with the continuity condition.\\vspace{0.2cm}\n\nThe following Theorems 1-2 are the main results of this section.\\vspace{0.1cm}\n\n{\\bf Theorem 1} Assume that the generator $g$ satisfies assumptions (H1) and (H3). Then for each $\\xi\\in\\LT$, BSDE (1) admits at most one solution $(y_\\cdot,z_\\cdot)$ such that $y_\\cdot$ belongs to the class (D) and $z_\\cdot$ belongs to $\\bigcup_{\\beta>\\alpha}\\M^\\beta$,\nwhich leads to that it admits at most one $L^1$ solution.\\vspace{0.1cm}\n\n{\\bf Proof}\\ \\ Assume that (H1) and (H3) hold and that both $(y_t,z_t)_{t\\in\\T}$ and $(y'_t,z'_t)_{t\\in\\T}$ are solutions of BSDE (1) such that both $(y_t)_{t\\in\\T}$ and $(y'_t)_{t\\in\\T}$ belong to the class (D), and both $(z_t)_{t\\in\\T}$ and $(z'_t)_{t\\in\\T}$ belong to $\\M^\\beta$ for some $\\beta\\in(\\alpha, 1)$.\\vspace{0.1cm}\n\nWe first show that $(y_t-y'_t)_{t\\in\\T}\\in \\s^{\\beta\/\\alpha}$. In fact, let us fix $n\\geq 1$ and denote $\\tau_n$ the stopping time\n$$\n\\tau_n:=\\inf\\left\\{t\\in\\T:\\int^t_{0}(|z_s|^2+|z'_s|^2)\n\\,\\mathrm{d} s\\geq n\\right\\}\\wedge T.\n$$\nCorollary 2.3 in \\citet{Bri03} leads to the following inequality with setting $\\hat{y}_\\cdot:=y_\\cdot-y'_\\cdot$ and   $\\hat{z}_\\cdot:=z_\\cdot-z'_\\cdot$, and $t\\in\\T$,\n\\begin{equation}\n\\Dis|\\hat{y}_{t\\wedge\\tau_n}|\\leq \\Dis |\\hat{y}_{\\tau_n}|+\\int^{\\tau_n}_{t\\wedge\\tau_n}\n\\left\\langle {\\hat{y}_s\\over |\\hat{y}_s| }\\mathbbm{1}_{|\\hat{y}_s|\\neq 0},g(s,y_s,z_s)-g(s,y'_s,z'_s)\\right\\rangle\\\n\\mathrm{d}s -\\int^{\\tau_n}_{t\\wedge\\tau_n}\\left\\langle {\\hat{y}_s\\over |\\hat{y}_s|}\\mathbbm{1}_{|\\hat{y}_s|\\neq 0},\\hat{z}_s\\mathrm{d}B_s\\right\\rangle.\n\\end{equation}\nIt follows from assumptions (H1) and (H3) that $\\ass$,\n\\begin{equation}\n\\begin{array}{ll}\n&\\Dis \\left\\langle{\\hat{y}_s\\over |\\hat{y}_s|} \\mathbbm{1}_{|\\hat{y}_s|\\neq 0},\ng(s,y_s,z_s)-g(s,y'_s,z'_s)\\right\\rangle\\vspace{0.1cm}\\\\\n\\leq &\\Dis \\left\\langle{\\hat{y}_s\\over |\\hat{y}_s|} \\mathbbm{1}_{|\\hat{y}_s|\\neq 0},\ng(s,y_s,z_s)-g(s,y'_s,z_s)\\right\\rangle\n+|g(s,y'_s,z_s)-g(s,y'_s,z'_s)|\\vspace{0.1cm}\\\\\n\\leq &\\Dis\\rho(|\\hat{y}_s|)\n+2\\gamma\\left(g_s+|y'_s|+|z'_s|+|z_s|\\right)^\\alpha.\n\\end{array}\n\\end{equation}\nThen, combining (2) with (3) we can deduce that for each $n\\geq 1$ and $t\\in\\T$,\n\\begin{equation}\n|\\hat{y}_{t\\wedge\\tau_n}|\\leq\\E\\left[|\\hat{y}_{\\tau_n}|\n+\\left.\\int^{\\tau_n}_{t\\wedge\\tau_n}\\rho(|\\hat{y}_s|)\\ \\mathrm{d}s\\right|\\F_t\\right]+G(t),\n\\end{equation}\nwhere\n$$\nG(t):=2\\gamma\\E\\left[\\left.\\int^{T}_{0}\\left(g_s+\n|y'_s|+|z'_s|+|z_s|\\right)^\\alpha\\ \\mathrm{d}s\\right|\\F_t\\right].\n$$\nFurthermore, since $y'_\\cdot$ belongs to the class (D), both $z_\\cdot$ and $z'_\\cdot$ belong to ${\\rm M}^\\beta$ with $\\beta>\\alpha$, and $\\E\\left[\\int_0^Tg_t\\ {\\rm d}t\\right]<+\\infty$, we can use Doob's inequality, H\\\"{o}lder's inequality and Jensen's inequality to obtain that\n\\begin{equation}\n\\E\\left[\\sup_{t\\in\\T}|G(t)|^{\\beta\/\\alpha}\\right]<+\\infty.\n\\end{equation}\nThus, since $\\hat y_\\cdot$ belongs to the class (D) and $\\rho(\\cdot)$ increases at most linearly, we can send $n$ to $+\\infty$ in (4) and use Lebesgue's dominated convergence theorem, in view of $\\tau_n\\To T$ as $n\\To \\infty$, $\\hat{y}_T=0$ and Remark 1, to get that for each $t\\in \\T$,\n$$\n|\\hat{y}_t|\\leq G(t)+\\E\\left[\\left.\\int_t^T\n\\rho(|\\hat{y}_s|)\\ \\mathrm{d}s\\right|\\F_t\\right]\\leq AT+G(t)+A\\int_t^T\\E\\left[\\left.\n|\\hat{y}_s|\\right|\\F_t\\right]\\mathrm{d}s,\n$$\nand then\n$$\n\\E\\left[\\left.|\\hat{y}_r|\\right|\\F_t\\right]\\leq AT+G(t)+A\\int_r^T\n\\E\\left[\\left.|\\hat{y}_s|\\right|\\F_t\\right]{\\rm d}s,\\quad r\\in [t,T].\\vspace{0.2cm}\n$$\nGronwall's inequality yields that $\\E\\left[\\left.|\\hat{y}_r|\\right|\\F_t\\right]\\leq\n\\left(AT+G(t)\\right)\\cdot e^{A(T-r)},\\ r\\in [t,T]$, form which, by letting $r=t$, we have\n\\begin{equation}\n|\\hat{y}_t|\\leq \\left(AT+G(t)\\right)\\cdot e^{AT}.\n\\end{equation}\nThis inequality together with (5) leads to\n\\begin{equation}\n\\hat y_\\cdot=(y_\\cdot-y'_\\cdot)\\in \\s^p\\ \\text{with}\\ p:=\\beta\/\\alpha>1.\n\\end{equation}\n\nIn the sequel, note that $(\\hat y_t, \\hat z_t)_{t\\in\\T}$ is a solution of the following BSDE:\n\\begin{equation}\n\\hat y_t=\\int^T_t\\hat g(s,\\hat y_s,\\hat z_s)\\ \\mathrm{d} s-\\int^T_t\\hat z_s\\,\\mathrm{d} B_s,\\quad t\\in\\T,\n\\end{equation}\nwhere for each $(y,z)\\in\\R^k\\times\\R^{k\\times d}$,\n$\\hat g(t,y,z):=g(t,y+y'_t,z+z'_t)\n-g(t,y'_t,z'_t)$.\nIt follows from assumptions (H1) and (H3) on $g$ together with Remark 1 that $\\as$, for each $(y,z)\\in \\R^k\\times\\R^{k\\times d}$,\n\\begin{equation}\n\\begin{array}{lll}\n\\left\\langle y,\\hat g(t,y,z)\\right\\rangle&\\leq&\\Dis \\left\\langle y, g(t,y+y'_t,z+z'_t)\n-g(t,y'_t,z+z'_t)\\right\\rangle+|y||g(t,y'_t,z+z'_t)-g(t,y'_t,z'_t)|\\\\\n&\\leq&\\Dis \\bar\\kappa(|y|^2)+\\lam |y||z|\\leq \\Dis A|y|^2+\\lam |y||z|+A,\n\\end{array}\n\\end{equation}\nand\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis |y|^{p-1}\\left\\langle {y\\over |y|}\\mathbbm{1}_{|y|\\neq 0},\\hat g(t,y,z)\\right\\rangle\n&\\leq&\\Dis |y|^{p-1}\\left\\langle {y\\over |y|}\\mathbbm{1}_{|y|\\neq 0}, g(t,y+y'_t,z+z'_t)\n-g(t,y'_t,z+z'_t)\\right\\rangle\\\\\n&&\\Dis +|y|^{p-1}|g(t,y'_t,z+z'_t)-g(t,y'_t,z'_t)|\n\\vspace{0.1cm} \\\\\n&\\leq&\\Dis \\kappa(|y|^p)+\\lam |y|^{p-1}|z|,\n\\end{array}\n\\end{equation}\nwhere the functions $\\bar\\kappa(\\cdot)$ and $\\kappa(\\cdot)$ are respectively defined in (H1a)$_2$ and (H1a)$_p$. Thus, on one hand, inequality (9) means that the generator $\\hat g$ of BSDE (8) satisfies assumption (A1) with $\\mu=A,\\ \\nu=\\lam,\\ f_t\\equiv 0\\ \\text{and}\\ \\varphi_t\\equiv A$. It then follows from Proposition 1 together with (7) that $(\\hat y_t, \\hat z_t)_{t\\in\\T}$ is an $L^p$ solution of BSDE (8). On the other hand, inequality (10) means that the generator $\\hat g$ of BSDE (8) also satisfies assumption (A2) with $\\psi(\\cdot)=\\kappa(\\cdot),\\ \\nu=\\lam\\ \\text{and}\\ f_t\\equiv 0$. It then follows from Proposition 2 with $u=0$ that there exists a positive constant $C_{\\lam,p,T}$ depending only on $\\lam$, $p$ and $T$ such that for each $t\\in \\T$,\\vspace{0.1cm}\n\\begin{equation}\n\\E\\left[\\sup_{s\\in [t,T]}|\\hat y_s|^p\\right]\\leq C_{\\lam,p,T} \\int_t^T\\kappa\\left(\\E\\left[|\\hat y_s|^p\\right]\\right)\\ {\\rm d}s\\leq C_{\\lam,p,T} \\int_t^T\\kappa\\left(\\E\\left[\\sup_{u\\in [s,T]}|\\hat y_u|^p\\right]\\right)\\ {\\rm d}s.\\vspace{0.1cm}\n\\end{equation}\nThus, in view of the fact that $\\int_{0^+} {{\\rm d}u\\over \\kappa(u)}=+\\infty$, Bihari's inequality (Lemma 2) yields that\n\\begin{equation}\n\\E\\left[\\sup_{t\\in [0,T]}|y_t-y'_t|^p\\right]=\\E\\left[\\sup_{t\\in [0,T]}|\\hat y_t|^p\\right]=0.\\vspace{0.2cm}\n\\end{equation}\n\nFinally, by (9), Remark 1 and Lemma 1 we can check that the generator $\\hat g$ of BSDE (8) satisfies assumption (A1) with\n$\\mu=m+2A,\\ \\nu=\\lam,\\ f_t\\equiv 0\\ {\\rm and}\\ \\ \\varphi_t=\\bar\\kappa({2A\\over m+2A})$\nfor each $m\\geq 1$. It then follows from Proposition 1 with $u=t=0$ that there exists a positive constant $C_{m,\\lam,p,T}$ depending on $m$, $\\lam$, $p$ and $T$, and a positive constant $C_{p}$ depending only on $p$ such that for each $m\\geq 1$,\n\\begin{equation}\n\\E\\left[\\left(\\int_0^T |\\hat z_s|^2\\ {\\rm\nd}s\\right)^{p\/2}\\right]\\leq C_{m,\\lam,p,T}\\E\\left[\\sup\\limits_{t\\in [0,T]}|\\hat y_t|^p\\right]+C_{p}\\left(\\bar\\kappa({2A\\over m+2A})\\cdot T\\right)^{p\/2}.\n\\end{equation}\nThus, in view of (12) and the fact that $\\bar\\kappa(\\cdot)$ is a continuous function with $\\bar\\kappa(0)=0$, sending $m\\To\\infty$ in the previous inequality we deduce that\n$$\n\\E\\left[\\left(\\int_0^T |z_s-z'_s|^2\\\n{\\rm d}s\\right)^{p\/2}\\right]=\\E\\left[\\left(\\int_0^T |\\hat z_s|^2\\ {\\rm\nd}s\\right)^{p\/2}\\right]=0.\n$$\nThe proof of Theorem 1 is then complete. \\vspace{0.3cm}\\hfill $\\Box$\n\nWe now turn to the existence part of our study. We will prove the following result.\\vspace{0.1cm}\n\n{\\bf Theorem 2} Assume that the generator $g$ satisfies assumptions (H1)-(H3). In the case when the $\\alpha$ defined in (H3) values in $[1\/2,1)$, we further assume that there exists a constant $\\bar p>1$ such that the function $\\rho(\\cdot)$ in (H1) satisfies\n\\begin{equation}\n\\int_{0^+}{u^{\\bar p-1}\\over \\rho^{\\bar p}(u)}{\\rm d}u=+\\infty.\\vspace{0.2cm}\n\\end{equation}\nThen for each $\\xi\\in\\LT$, BSDE (1) admits an $L^1$ solution $(y_t,z_t)_{t\\in\\T}$.\\vspace{0.3cm}\n\n{\\bf Remark 2} It follows from Proposition 1 in \\citet{Fan15} that if $g$ satisfies (H1) with a $\\rho(\\cdot)$ satisfying (14) for some $\\bar p>1$, then $g$ must satisfy (H1b)$_{\\bar p}$. And vice versa. This fact will be perfectly utilized later.\\vspace{0.2cm}\n\nThe following Proposition 5 is the first step to  prove Theorem 2, which studies the case where the generator $g$ does not depend on the variable $z$.\\vspace{0.2cm}\n\n{\\bf Proposition 5} Let the generator $g$ be independent of $z$ and satisfy assumptions (H1)-(H2). Then for each $\\xi\\in\\LT$, BSDE (1) admits an $L^1$ solution $(y_t,z_t)_{t\\in\\T}$.\\vspace{0.2cm}\n\n{\\bf Proof}\\ \\ Assume that $\\xi\\in\\LT$, $g$ is independent of $z$ and assumptions (H1) and (H2) hold. For each $n\\geq 1$, we denote $q_n(x)=xn\/(n\\vee |x|)$ and set\n$$\n\\xi^n:=q_n(\\xi),\\quad   g^n(t,y):=g(t,y)-g(t,0)+q_n(g(t,0)).\n$$\nNote that both $|\\xi^n|$ and $|g^n(t,0)|$ are bounded by $n$, and that the generator $g^n(t,y)$ satisfies (H1) and  (H2) for each $n\\geq 1$. It then follows from\nCorollary 2 with $p=2$ in \\citet{Fan15} that the following BSDE\n\\begin{equation}\ny^n_t=\\xi^n+\\int^{T}_{t}g^n(s,y^n_s)\\,\\mathrm{d} s-\\int^{T}_{t}z_s^n\\,\\mathrm{d} B_s,\\quad t\\in\\T\n\\end{equation}\nadmits a unique $L^2$ solution $(y^n_t,z^n_t)_{t\\in\\T}$.\\vspace{0.2cm}\n\nFor each $n,i\\geq 1$, we set $\\hat{y}^{n,i}_\\cdot:=y^{n+i}_\\cdot-y^n_\\cdot$, $\\hat z^{n,i}_\\cdot:=z^{n+i}_\\cdot-z^n_\\cdot$ and $\\hat\\xi^{n,i}:=\\xi^{n+i}-\\xi^n$. It follows from Corollary 2.3 in \\cite{Bri03} that for each $n,i\\geq 1$ and $t\\in\\T$,\n\\begin{equation}\n|\\hat{y}^{n,i}_t|\\leq \\Dis |\\hat{\\xi}^{n,i}|+\\int^{T}_{t}\n\\left\\langle {\\hat y^{n,i}_s\\over |\\hat y^{n,i}_s|}\\mathbbm{1}_{|\\hat y^{n,i}_s|\\neq 0}, g^{n+i}(s,y^{n+i}_s)-g^n(s,y^n_s)\\right\\rangle\n\\ \\mathrm{d} s\\Dis -\\int^{T}_{t}\\left\\langle {\\hat y^{n,i}_s\\over |\\hat y^{n,i}_s|}\\mathbbm{1}_{|\\hat y^{n,i}_s|\\neq 0},\\hat z^{n,i}_s\\,\\mathrm{d} B_s\\right\\rangle.\n\\end{equation}\nIt follows from assumption (H1) and definition of $g^n(t,y)$ that $\\ass$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\left\\langle {\\hat y^{n,i}_s\\over |\\hat y^{n,i}_s|}\\mathbbm{1}_{|\\hat y^{n,i}_s|\\neq 0}, g^{n+i}(s,y^{n+i}_s)-g^n(s,y^n_s)\\right\\rangle\n&\\leq &\\Dis \\left\\langle {\\hat y^{n,i}_s\\over |\\hat y^{n,i}_s|}\\mathbbm{1}_{|\\hat y^{n,i}_s|\\neq 0}, g^{n+i}(s,y^{n+i}_s)-g^{n+i}(s,y^n_s)\\right\\rangle\n\\vspace{0.1cm}\\\\\n&&+|g^{n+i}(s,y^n_s)-g^n(s,y^n_s)|\\vspace{0.1cm} \\\\\n&\\leq &\\Dis \\rho(|\\hat y^{n,i}_s|)+|g(s,0)|       \\mathbbm{1}_{|g(s,0)|>n}.\n\\end{array}\n\\end{equation}\nThen, combining (16) and (17), in view of Fubini's Theorem and Jensen's inequality, we can get that for each $n,i\\geq 1$ and $t\\in\\T$,\n\\begin{equation}\n|\\hat y_t^{n,i}|\\leq H_n(t)+\\int^{T}_{t}\\rho\\left(\\E\\left[\\left.|\\hat y^{n,i}_s|\\right|\\F_t\\right]\\right)\\mathrm{d}s,\n\\end{equation}\nwhere\n$$H_n(t):=\\E\\left[|\\xi|\\mathbbm{1}_{|\\xi|>n} +\\left.\\int^{T}_{0}|g(s,0)|\\mathbbm{1}_{|g(s,0)|>n}\n\\mathrm{d}s\\right|\\F_t\\right].\\vspace{0.1cm}\n$$\n\nIn the sequel, by virtue of Fubini's theorem and Jensen's inequality, it follows from (18) that for each $n,i\\geq 1$ and $t\\in\\T$,\n$$\n\\E\\left[\\left.|\\hat y_r^{n,i}|\\right|\\F_t\\right]\\leq H_n(t)+\\int^{T}_{r}\\rho\\left(\\E\\left[\\left.|\\hat y^{n,i}_s|\\right|\\F_t\\right]\\right)\\mathrm{d}s,\\quad r\\in [t,T].\n$$\nThen in view of Remark 1, Gronwall's inequality yields that for each $n,i\\geq 1$ and $(\\omega,t)\\in \\Omega\\times\\T$,\n\\begin{equation}\n|\\hat{y}_t^{n,i}|=\\E\\left[\\left.|\\hat{y}_{t}^{n,i}|\n\\right|\\F_t\\right]\\leq \\left(H_n(t)+AT\\right)\\cdot e^{A(T-t)}.\n\\end{equation}\nThis inequality together with Lemma 6.1 in \\cite{Bri03} leads to that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis\\sup\\limits_{n\\geq 1}\\E\\left[\\sup\\limits_{i\\geq 1}\\sup\\limits_{t\\in\\T} |\\hat{y}_t^{n,i}|^\\beta\\right]&\\leq &\\Dis C\\left(1+\\sup\\limits_{n\\geq 1} \\E\\left[\\sup\\limits_{t\\in\\T} |H_n(t)|^\\beta\\right]\\right)\\vspace{0.1cm}\\\\\n&\\leq & \\Dis C\\left(1+{1\\over 1-\\beta}\\sup\\limits_{n\\geq 1}(\\E[H_n(T)])^\\beta\\right)<+\\infty,\n\\end{array}\n\\end{equation}\nwhere $C>0$ is a constant depending only on $A,T$ and $\\beta$. This means that the sequence\n$$\\left\\{\\sup_{i\\geq 1}\\sup_{t\\in\\T} |\\hat{y}_t^{n,i}|^{\\beta'}\\right\\}_{n=1}^{+\\infty}\n$$\nis uniformly integrable for each $\\beta'\\in (0,1)$.\\vspace{0.1cm}\n\nOn the other hand, by virtue of Fubini's theorem and Jensen's inequality, it follows from (18) that for each $n,i\\geq 1$ and $t\\in\\T$,\n$$\n\\E\\left[\\left.|\\hat{y}_{t}^{n,i}|\\right|\\F_u\\right]\\leq\nH_n(u)+\\int_t^T\\rho\\left(\\E\\left[\\left.|\\hat{y}_{s}^{n,i}|\n\\right|\\F_u \\right]\\right){\\rm d}s,\\ \\ u\\in[0,t],\n$$\nand then, in view of the fact that $\\rho(\\cdot)$ is a nondecreasing function,\n\\begin{equation}\n\\Dis h_n(t)\\leq \\sup_{u\\in\\T}H_n(u)+\\int_t^T\\rho\\left(h_n(s)\\right){\\rm d}s,\\ \\ t\\in\\T,\n\\end{equation}\nwhere\n$$\nh_n(t):=\\sup_{i\\geq 1}\\sup\\limits_{0\\leq u\\leq t}\n\\E\\left[\\left.|\\hat{y}_{t}^{n,i}|\n\\right|\\F_u\\right]\\vspace{0.2cm}\n$$\nare all nonnegative functions. Thus, by virtue of  Lemma 2 we can deduce that for each $n\\geq 1$ and $\\omega\\in \\Omega$,\n\\begin{equation}\nh_n(t)\\leq \\Theta^{-1}(\\Theta(\\sup_{t\\in\\T}H_n(t))+T),\\ \\ t\\in\\T,\n\\end{equation}\nwhere\n\\begin{equation}\n\\Theta(x):=\\int_1^x {1\\over \\rho(x)}{\\rm d}x,\\ x>0 \\vspace{0.2cm}\n\\end{equation}\nis a strictly increasing and continuous function valued in $\\R$, and $\\Theta^{-1}$ is the inverse function of $\\Theta$. Furthermore, by the maximum inequality with respect to sub-martingale and Lebesgue's dominated convergence theorem we have, for each $\\epsilon>0$, as $n\\To \\infty$,\n\\begin{equation}\n\\mathbbm{P}\\left(\\left\\{\\sup_{t\\in\\T} |H_n(t)|\\geq \\epsilon\\right\\}\\right)\\leq {1\\over \\epsilon}\\E[|H_n(T)|]\\longrightarrow 0.\n\\end{equation}\nThus, noticing by the definition of $h_n(t)$ together with (22) that\n$$\n\\sup_{t\\in\\T}\\sup_{i\\geq 1} \\E\\left[|\\hat{y}_{t}^{n,i}|\\right]\n\\leq \\sup_{t\\in\\T} h_n(t)\\leq \\Theta^{-1}(\\Theta(\\sup_{t\\in\\T}H_n(t))+T)\n$$\nand\n$$\n\\sup_{t\\in\\T}\\sup_{i\\geq 1} |\\hat{y}_{t}^{n,i}|=\\sup_{t\\in\\T}\\sup_{i\\geq 1}\\E\\left[\\left.|\\hat{y}_{t}^{n,i}|\\right|\\F_t\\right]\n\\leq \\sup_{t\\in\\T} h_n(t)\\leq \\Theta^{-1}(\\Theta(\\sup_{t\\in\\T}H_n(t))+T),\n\\vspace{0.2cm}\n$$\nfrom (24) we can deduce that for each $\\epsilon>0$,\n$$\n0\\leq \\Lim \\mathbbm{P}\\left(\\left\\{\\sup_{i\\geq 1} \\sup_{t\\in\\T}\\E\\left[|\\hat{y}_{t}^{n,i}|\\right]\\geq \\epsilon\\right\\}\\right)\\leq\n\\Lim \\mathbbm{P}\\left(\\left\\{ \\sup_{t\\in\\T}|H_n(t)|\\geq \\Theta^{-1}(\\Theta(\\epsilon)-T)\\right\\}\\right)=0\n$$\nand\n\\begin{equation}\n0\\leq \\Lim \\mathbbm{P}\\left(\\left\\{\\sup_{i\\geq 1}\\sup_{t\\in\\T} |\\hat{y}_{t}^{n,i}|\\geq \\epsilon\\right\\}\\right)\\leq\n\\Lim \\mathbbm{P}\\left(\\left\\{ \\sup_{t\\in\\T}|H_n(t)|\\geq \\Theta^{-1}(\\Theta(\\epsilon)-T)\\right\\}\\right)=0.\n\\end{equation}\nThat is to say,\n\\begin{equation}\n\\Lim \\sup_{i\\geq 1} \\sup_{t\\in\\T} \\E\\left[|\\hat{y}_{t}^{n,i}|\\right]=0\n\\end{equation}\nand $\\left\\{\\sup_{i\\geq 1}\\sup_{t\\in\\T} |\\hat{y}_{t}^{n,i}|\\right\\}_{n=1}^{+\\infty}$\nconverges to $0$ in probability as $n\\To\\infty$.\\vspace{0.2cm}\n\nCombining (20) and (25) we get that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\Lim \\sup_{i\\geq 1}\\E\\left[\\sup_{t\\in\\T} |\\hat{y}_{t}^{n,i}|^\\beta\\right]=0.\n\\end{equation}\nIt then follows from (19), (26) and (27) that there exists a process $(y_t)_{t\\in \\T}$ which belongs to the class (D) and the space $\\bigcap_{\\beta\\in (0,1)}\\s^\\beta$ such that\n\\begin{equation}\n\\Lim \\sup_{t\\in\\T}\\E\\left[|y_t^{n}-y_t|\\right]=0\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\Lim \\E\\left[\\sup_{t\\in\\T} |y_t^{n}-y_t|^\\beta\\right]=0.\\vspace{0.2cm}\n\\end{equation}\n\nFurthermore, note that $(\\hat y_t^{n,i},\\hat z_t^{n,i})_{t\\in\\T}$ is a solution of the following BSDE:\n\\begin{equation}\n\\hat{y}_t^{n,i}=\\hat \\xi^{n,i}+\\int^{T}_{t}\\hat g^{n,i}(s,\\hat y_s^{n,i})\\ \\mathrm{d} s-\\int^{T}_{t}\\hat{z}_s^{n,i}\\ \\mathrm{d}B_s, \\quad t\\in\\T,\n\\end{equation}\nwhere for each $n,i\\geq 1$ and $y\\in\\R^k$,\n$\n\\hat g^{n,i}(t,y):=g^{n+i}(t,y+y^n_t)-g^n(t,y^n_t).\n$\nIt follows from assumption (H1) on $g$ together with Remark 1 that $\\as$, for each $n,i\\geq 1$ and $y\\in \\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\hat g^{n,i}(t,y)\\rangle&\\leq &\\Dis \\langle y,g^{n+i}(t,y+y^n_t)-g^{n+i}(t,y^n_t)\\rangle\n+|y||g^{n+i}(t,y^n_t)-g^n(t,y^n_t)|\\vspace{0.1cm}\\\\\n&\\leq &\\Dis \\kappa(|y|^2)+|y||g(t,0)|\\mathbbm{1}_{|g(t,0)|>n},\n\\end{array}\n\\end{equation}\nwhere the function $\\kappa(\\cdot)$ is defined in (H1a)$_2$. Thus, by (31), Remark 1 and Lemma 1 we can check that the generator $\\hat g^{n,i}$ of BSDE (30) satisfies assumption (A1) with\n$\np=\\beta,\\ \\mu=m+2A,\\ \\nu=0,\\ f_t=|g(t,0)|\\mathbbm{1}_{|g(t,0)|>n}\\ {\\rm and}\\ \\ \\varphi_t=\\kappa({2A\\over m+2A})\n$\nfor each $m\\geq 1$ and $\\beta\\in (0,1)$. It then follows from Proposition 1 with $u=t=0$ and $p=\\beta$ that for each $m,n,i\\geq 1$, there exist a constant $C_{m,A,\\beta,T}>0$ depending only on $m, A,\\beta$ and $T$, and a constant $C>0$ such that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\E\\left[\\left(\\int_0^T |\\hat z_s^{n,i}|^2\\ {\\rm\nd}s\\right)^{\\beta\/2}\\right]&\\leq &\\Dis C_{m,A,\\beta,T}\\E\\left[\\sup\\limits_{t\\in [0,T]}|\\hat y_t^{n,i}|^\\beta\\right]+\nC\\left(\\kappa({2A\\over m+2A})\\cdot T\\right)^{\\beta\/2}\\vspace{0.1cm}\\\\\n&&\\Dis +C\\E\\left[\\left(\\int^{T}_{0}\n|g(t,0)|\\mathbbm{1}_{|g(t,0)|>n}\\mathrm{d}t\\right)^\\beta\n\\right].\n\\end{array}\n\\end{equation}\nThus, taking superemum with respect to $i$ and sending first $n\\To\\infty$ ($m$ being fixed) and then $m\\To\\infty$ in (32), by virtue of (27), (H2), Lebesgue's dominated convergence theorem and the fact that $\\kappa(\\cdot)$ is continuous function with $\\kappa(0)=0$, we can deduce that for each $\\beta\\in (0,1)$,\n$$\n\\Lim \\sup\\limits_{i\\geq 1}\\E\\left[\\left(\\int_0^T |z^{n+i}_s-z^{n}_s|^2\\\n{\\rm d}s\\right)^{\\beta\/2}\\right]=\\Lim \\sup\\limits_{i\\geq 1}\\E\\left[\\left(\\int_0^T |\\hat z^{n,i}_s|^2\\ {\\rm\nd}s\\right)^{\\beta\/2}\\right]=0,\n$$\nwhich means that there exists a process $(z_t)_{t\\in \\T}$ which belongs to $\\bigcap_{\\beta\\in (0,1)}\\s^\\beta$ such that\\vspace{0.1cm}\n\\begin{equation}\n\\RE\\ \\beta\\in (0,1),\\ \\ \\Lim \\E\\left[\\left(\\int_0^T |z^n_s-z_s|^2\\\n{\\rm d}s\\right)^{\\beta\/2}\\right]=0.\n\\end{equation}\n\nFinally, since $\\int^{T}_{t}z^n_s\\ \\mathrm{d} B_s$ converges to $\\int^{T}_{t}z_s\\ \\mathrm{d} B_s$ under the uniform convergence in probability (ucp for short) by (33) and since $y\\mapsto g(t,y)$ is continuous and (29) holds true, we can easily check by taking limit in both sides of BSDE (15) under ucp that $(y_t,z_t)_{t\\in\\T}$ is an $L^1$ solution of BSDE (1). \\vspace{0.2cm}\\hfill $\\Box$\n\n{\\bf Remark 3}\\ Compared with the existing works, the argument between (18)-(29) involved in Gronwall's inequality, Bihari's ineqality and the relationship between convergence in $L^1$ and in probability seems to be  new. \\vspace{0.2cm}\n\nWith Proposition 5 in the hand we can prove the main existence result.\\vspace{0.2cm}\n\n{\\bf Proof of Theorem 2}\\ \\ Assume that $\\xi\\in\\LT$ and the generator $g$ satisfies assumptions (H1)-(H3). Furthermore, in the case of $\\alpha\\in [1\/2,1)$ in (H3) we also assume that $g$ satisfies (H1) with a function $\\rho(\\cdot)$ satisfying (14) for some $\\bar p>1$. We will use some kind of Picard's iterative procedure. \\vspace{0.1cm}\n\nLet us set $(y^0_\\cdot,z^0_\\cdot):=(0,0)$. Note by (H3) that $g$ has a sublinear growth with respect to $z$. By virtue of H\\\"{o}lder's inequality and assumptions (H1) and (H2) of $g$, it is not hard to verify that the generator $g(t,y,z_t)$ satisfies (H1) and (H2) for each $(z_t)_{t\\in\\T}\\in\\M^1$. Thus, with the help of Proposition 5, we can define the process sequence $\\{(y^n_t,z^n_t)_{t\\in\\T}\\}_{n=1}^\\infty$ recursively,\n\\begin{equation}\ny^{n}_t=\\xi+\\int^{T}_{t}g(s,y^{n}_s,z^{n-1}_s)\\,\\mathrm{d} s-\\int^{T}_{t}z^{n}_s\\,\\mathrm{d} B_s, \\quad t\\in\\T,\n\\end{equation}\nwhere for each $n\\geq 1$, $(y^n_t,z^n_t)_{t\\in\\T}$ belongs to the space $\\s^\\beta\\times \\M^\\beta$ for each $\\beta\\in (0,1)$, and $(y^n_t)_{t\\in\\T}$ belongs to the class (D).\\vspace{0.2cm}\n\nFor each $n,i\\geq 0$, set $\\hat{y}^{n,i}_\\cdot:=y^{n+i}_\\cdot-y^n_\\cdot$ and $\\hat z^{n,i}_\\cdot:=z^{n+i}_\\cdot-z^n_\\cdot$. Arguing as in the proof of Theorem 1, we can get, in view of (H1) and (H3), that for each $n,i\\geq 1$,\n$$|\\hat{y}^{n,i}_t|\\leq \\left(AT+G^{n,i}(t)\\right)\\cdot e^{AT},\\quad t\\in\\T,$$\nwhere\n\\begin{equation}\nG^{n,i}(t):=2\\gamma\\E\\left[\\left.\\int^{T}_{0}\\left(g_s+\n|y^n_s|+|z^{n-1}_s|+|z^{n+i-1}_s|\\right)^\\alpha\\ \\mathrm{d}s\\right|\\F_t\\right]\\in \\s^q\n\\end{equation}\nas soon as $\\alpha q<1$ with $q>1$. Hence, for each $n,i\\geq 1$, we know that $(\\hat y_t^{n,i})_{t\\in\\T}$ belongs to the space $\\s^q$ as soon as $\\alpha q<1$ with $q>1$. In the sequel, we will deal with two cases respectively:\n$$\n(i)\\ \\ \\alpha\\in (0,1\/2); \\quad \\quad (ii)\\ \\ \\alpha\\in [1\/2,1).\n$$\n\n{\\bf Case $(i)$}: In this case, we can pick $q=2$, then for each $n,i\\geq 1$,\n\\begin{equation}\n(\\hat y_t^{n,i})_{t\\in\\T}\\in \\s^2.\n\\end{equation}\nNote that for each $n,i\\geq 1$, $(\\hat y_t^{n,i},\\hat z_t^{n,i})_{t\\in\\T}$ is a solution of the following BSDE:\n\\begin{equation}\n\\hat{y}_t^{n,i}=\\int_t^T \\bar{g}^{n,i}(s,\\hat{y}_s^{n,i})\\ {\\rm\nd}s-\\int_t^T \\hat{z}_s^{n,i}{\\rm d}B_s,\\ \\ \\ t\\in \\T,\n\\end{equation}\nwhere for each $y\\in \\R^k$, $\\bar{g}^{n,i}(s,y):=g(s,y+y_s^n,z^{n+i-1}_s)-\ng(s,y_s^n,z^{n-1}_s)$. It follows from (H1), (H3) and Remark 1 that $\\as$, for each $y\\in\\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\bar{g}^{n,i}(t,y)\\rangle &\\leq &\\Dis \\kappa(|y|^2)+2\\gamma|y|\\left(g_s+\n|y^n_s|+|z^{n-1}_s|+|z^{n+i-1}_s|\\right)^\\alpha\\\\\n&\\leq & \\Dis A|y|^2+2\\gamma|y|\\left(g_s+\n|y^n_s|+|z^{n-1}_s|+|z^{n+i-1}_s|\\right)^\\alpha+A,\n\\end{array}\n\\end{equation}\nand\n\\begin{equation}\n\\langle y,\\bar{g}^{n,i}(t,y)\\rangle \\leq \\kappa(|y|^2)+\\lam\n|y||z^{n+i-1}_t-z^{n-1}_t|=\\kappa(|y|^2)+ \\lam |y||\\hat\nz^{n-1,i}_t|,\\vspace{0.2cm}\n\\end{equation}\nwhere $\\kappa(\\cdot)$ is defined in (H1a)$_2$. Thus, by (35) and (38) we know that the generator $\\bar g^{n,i}(t,y)$ of BSDE (37) satisfies assumption (A1) with\n$p=2,\\ \\mu=A,\\ \\nu=0,\\ f_t=2\\gamma\\left(g_t+\n|y^n_t|+|z^{n-1}_t|+|z^{n+i-1}_t|\\right)^\\alpha\\ {\\rm and}\\ \\varphi_t\\equiv A$.\nThen, in view of (36), it follows from Proposition 1 with $p=2$ that $\\hat z^{n,i}_\\cdot\\in {\\rm M}^2$. Consequently, for each $n,i\\geq 1$, $(\\hat y^{n,i}_t,\\hat z^{n,i}_t)_{t\\in \\T}$ is an $L^2$ solution of BSDE (37).\\vspace{0.2cm}\n\nOn the other hand, it follows from (39) that for each $n\\geq 2$ and $i\\geq 1$, the generator $\\bar g^{n,i}(t,y)$ of BSDE (37) also satisfies assumption (A2) with\n$\np=2,\\ \\psi(u)=\\kappa(u),\\ \\nu=0,\\ f_t=\\lam |\\hat z^{n-1,i}_t|.\n$\nThen, by Proposition 3 with $u=0$ and H\\\"{o}lder's inequality we can deduce that there exists a constant $C>0$ such that for each $t\\in [0,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n&&\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\hat\ny^{n,i}_r|^2\\right]+\\E\\left[\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right]\\vspace{0.1cm}\\\\\n&\\leq &\\Dis e^{C(T-t)}\\left\\{\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\hat y^{n,i}_r|^2 \\right]\\right){\\rm d}s+\\lam^2 (T-t)\\E\\left[\\int_t^T |\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right]\\right\\}.\n\\end{array}\n\\end{equation}\nNow, let\n$$\n\\delta T:=\\min\\left\\{{\\ln 2\\over C}, {1\\over 16\\lam^2}, {\\ln 2\\over 2A}\\right\\}\\ \\ {\\rm and}\\ \\\nT_j:=(T-j\\delta T)\\vee 0,\\ \\ \\RE\\ j=1,2,\\cdots\\vspace{0.2cm}\n$$\nThen for each $t\\in [T_1,T]$, we have\n\\begin{equation}\ne^{C(T-t)}\\leq 2, \\ \\ \\lam^2e^{C(T-t)}(T-t)\\leq {1\\over 8},\\ \\ e^{2A(T-t)}\\leq 2.\n\\end{equation}\nCombining (40) with (41) yields that for each $n\\geq 2$, $i\\geq 1$ and $t\\in [T_1,T]$,\\vspace{0.2cm}\n\\begin{equation}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\hat\ny^{n,i}_r|^2\\right]+\\E\\left[\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right]\n\\leq \\Dis 2\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\hat y^{n,i}_r|^2 \\right]\\right){\\rm d}s+{1\\over 8}\\E\\left[\\int_t^T |\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right].\\vspace{0.2cm}\n\\end{equation}\n\nFurthermore, note by Remark 1 that $\\kappa(x)\\leq A(x+1)$ for each $x\\geq 0$. Gronwall's inequality with (42) and (41) yields that for each $n\\geq 2$, $i\\geq 1$  and $t\\in [T_1,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\hat y^{n,i}_r|^2\\right]+\\E\\left[\\int_t^T | \\hat z^{n,i}_s|^2\\ {\\rm\nd}s\\right]&\\leq &\\Dis \\left(2AT+{1\\over 8}\\E\\left[\\int_t^T |\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right]\\right)\\cdot e^{2A(T-t)}\\vspace{0.1cm}\\\\\n&\\leq &\\Dis 4AT+{1\\over 4}\\E\\left[\\int_t^T\n|\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right].\n\\end{array}\n\\end{equation}\nBy picking $n=2$ and $i=m-2$ in (43) we get that for each $t\\in [T_1,T]$ and $m\\geq 3$,\n$$\n\\begin{array}{lll}\n\\Dis \\E\\left[\\int_t^T | z^m_s-z^2_s|^2\\ {\\rm\nd}s\\right]&\\leq &\\Dis 4AT+{1\\over 4}\\E\\left[\\int_t^T | z^{m-1}_s-z^1_s|^2\\ {\\rm d}s\\right]\\vspace{0.1cm}\\\\\n&\\leq &\\Dis 4AT+{1\\over 2}\\E\\left[\\int_0^T |z^2_s- z^1_s|^2\\ {\\rm d}s\\right]+{1\\over 2}\\E\\left[\\int_t^T | z^{m-1}_s-z^2_s|^2\\ {\\rm d}s\\right],\n\\end{array}\n$$\nfrom which we can obtain by induction that for each $t\\in [T_1,T]$,\n\\begin{equation}\n\\sup_{m\\geq 3} \\E\\left[\\int_t^T | z^m_s-z^2_s|^2\\ {\\rm\nd}s\\right]\\leq 8AT+\\E\\left[\\int_0^T |z^2_s-z^1_s|^2\\ {\\rm d}s\\right]<+\\infty.\n\\end{equation}\nIn addition, for each $n\\geq 2$, $i\\geq 1$ and $t\\in \\T$ we have\\vspace{0.1cm}\n\\begin{equation}\n\\Dis{1\\over 4}\\E\\left[\\int_t^T\n|\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right]\\leq \\Dis {1\\over 2}\\E\\left[\\int_t^T\n\\left(|z^{n-1+i}_s-z^2_s|^2+|z^{n-1}_s-z^2_s|^2\\right)\\ {\\rm d}s\\right]\\leq \\Dis \\sup_{m\\geq 1} \\E\\left[\\int_t^T | z^m_s-z^2_s|^2\\ {\\rm d}s\\right]\\vspace{-0.1cm}.\n\\end{equation}\nCombining (43), (45) and (44) yields that for each $t\\in [T_1,T]$,\n\\begin{equation}\n\\Dis\\sup_{n\\geq 2}\\sup_{i\\geq 1}\\left(\\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\hat y^{n,i}_r|^2\\right]+\\E\\left[\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right]\\right)\\leq \\Dis 12AT+\\E\\left[\\int_0^T |z^2_s- z^1_s|^2\\ {\\rm d}s\\right]<+\\infty.\\vspace{0.1cm}\n\\end{equation}\n\nNow, in view of (46), by first taking supremum with respect to $i$ and then taking $\\limsup$ with respect to $n$ in (42) and finally using Fatou's lemma, the monotonicity and continuity of the function $\\kappa(\\cdot)$ together with Bihari's inequality, we can deduce the existence of processes $(Y_t,Z_t)_{t\\in [T_1,T]}\\in {\\s}^2(T_1,T;\\R^k)\\times {\\rm M}^2(T_1,T;\\R^{k\\times d})$ such that\n\\begin{equation}\n\\Lim \\E\\left[\\sup_{t\\in [T_1,T]}|(y^n_t-y^1_t)-Y_t|^2+\\int_{T_1}^T\n|(z^n_t-z^1_t)-Z_t|^2\\ \\mathrm{d}t\\right]=0.\n\\end{equation}\nThus, note that $(y^1_t,z^1_t)_{t\\in\\T}\\in \\s^\\beta\\times \\M^\\beta$ for each $\\beta\\in (0,1)$ and $(y^1_t)_{t\\in\\T}$ belongs to the class (D). By passing to the limit in ucp for BSDE (34), in view of (47), (H2), (H3) and Lebesgue's dominated convergence theorem, we deduce that\n$\n(y_t,z_t)_{t\\in [T_1,T]}:=(Y_t+y^1_t,Z_t+z^1_t)_{t\\in [T_1,T]}\n$\nis an $L^1$ solution to the\nBSDE with parameters $(\\xi,T,g)$ on $[T_1,T]$.\\vspace{0.1cm}\n\nFinally, noticing that the $\\delta T>0$ depends only on $\\lam$ and $A$, we can find a minimal integer $N\\geq 1$ such that $T_N=0$. Thus, we can repeat, in finite steps, the above procedure to obtain an $L^1$ solution to BSDE (1) on $[T_2,T_1]$, $[T_3,T_2]$, $\\cdots$, $[0,T_{N-1}]$, and then we find an $L^1$ solution to BSDE (1) on $[0,T]$.\\vspace{0.2cm}\n\n{\\bf Case $(ii)$}: In this case, we can pick a $q\\in (1,\\bar p\\wedge {1\\over \\alpha})$, then for each $n,i\\geq 1$,\n\\begin{equation}\n(\\hat y_t^{n,i})_{t\\in\\T}\\in \\s^q.\n\\end{equation}\nNote that $q<\\bar p$ and that we also assume that equality (14) holds in this case. It follows from Proposition 1 in \\citet{Fan15} that $g$ also satisfies (H1b)$_{\\bar p}$ and then (H1b)$_q$. Note further that (37) and (38) is also true. In the same way as that in case $(i)$, it follows from Proposition 1 with $p=q$ that, in view of (48) and (35), $\\hat z^{n,i}_t\\in {\\rm M}^q$. Consequently, for each $n,i\\geq 1$, $(\\hat y^{n,i}_t,\\hat z^{n,i}_t)_{t\\in \\T}$ is an $L^q$ solution of BSDE (37).\\vspace{0.2cm}\n\nFurthermore, it follows from (H1b)$_q$ and (H3) of the generator $g$ together with Remark 1 that $\\as$, for each $y\\in \\R^k$,\\vspace{0.2cm}\n$$\n\\left\\langle {y\\over |y|}\\mathbbm{1}_{|y|\\neq 0}, \\bar{g}^{n,i}(t,y)\\right\\rangle \\leq \\varrho^{1\\over q}(|y|^q)+\\lam |\\hat z^{n-1,i}_t|,\n$$\nwhere $\\varrho(\\cdot)$ is defined in (H1b)$_q$. Then, for each $n\\geq 2$ and $i\\geq 1$, the generator $\\bar g^{n,i}(t,y)$ of BSDE (37) satisfies assumption (A3) with\n$\np=q,\\ \\phi(u)=\\varrho(u),\\ \\nu=0,\\ f_t=\\lam |\\hat z^{n-1,i}_t|.\n$\nThen, by Proposition 4 with $u=0$ and $p=q$ together with H\\\"{o}lder's inequality we can deduce that there exists a constant $C_q>0$ depending only on $q$ such that for each $n\\geq 2$, $i\\geq 1$ and $t\\in [0,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n&& \\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\hat\ny^{n,i}_r|^q\\right]+\\E\\left[\\left(\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]\\\\\n&\\leq &\\Dis e^{C_q(T-t)}\\left\\{\\int_t^T\\varrho\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\hat y^{n,i}_r|^q \\right]\\right){\\rm d}s+\\lam^q (T-t)^{q\/2}\\E\\left[\\left(\\int_t^T |\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]\\right\\}.\n\\end{array}\n\\end{equation}\nNow, let\n$$\n\\overline{\\delta T}:=\\min\\left\\{{\\ln 2\\over C_q}, \\left({1\\over 16\\lam^q}\\right)^{2\\over q},{\\ln 2\\over 2A}\\right\\}\\ \\ {\\rm and}\\ \\ \\bar T_j:=(T-j\\overline{\\delta T})\\vee 0,\\ \\ \\RE\\ j=1,2,\\cdots\\vspace{0.2cm}\n$$\nThen for each $t\\in [\\bar T_1,T]$, we have\n\\begin{equation}\ne^{C_q(T-t)}\\leq 2, \\ \\ \\lam^q e^{C_q(T-t)}(T-t)^{q\/2}\\leq {1\\over 8},\\ \\ e^{2A(T-t)}\\leq 2.\n\\end{equation}\nCombining (49) with (50) yields that for each $n\\geq 2$, $i\\geq 1$ and $t\\in [\\bar T_1,T]$,\n\\begin{equation}\n\\begin{array}{ll}\n& \\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\hat\ny^{n,i}_r|^q\\right]+\\E\\left[\\left(\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]\\vspace{0.1cm}\\\\\n\\leq &\\Dis 2\\int_t^T\\varrho\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\hat y^{n,i}_r|^q \\right]\\right){\\rm d}s+{1\\over 8}\\E\\left[\\left(\\int_t^T |\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right].\n\\end{array}\n\\end{equation}\n\nFurthermore, note by Remark 1 that $\\varrho(x)\\leq A(x+1)$ for each $x\\geq 0$. Gronwall's inequality with (51) and (50) yields that for each $n\\geq 2$, $i\\geq 1$ and $t\\in [\\bar T_1,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\hat y^{n,i}_r|^q\\right]+\\E\\left[\\left(\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]\n&\\leq &\\Dis \\left(2AT+{1\\over 8}\\E\\left[\\left(\\int_t^T | \\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]\\right)\\cdot e^{2A(T-t)}\\\\\n&\\leq &\\Dis 4AT+{1\\over 4}\\E\\left[\\left(\\int_t^T | \\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right],\\vspace{-0.2cm}\n\\end{array}\n\\end{equation}\nfrom which, in view of $q\\in (1,2)$ and the basic inequality\n$$\\left(\\int_t^T(a_s+b_s)^2\\ \\mathrm{d}s\\right)^{q\/2}\\leq 2\\left[\\left(\\int_t^T a^2_s\\ \\mathrm{d}s\\right)^{q\/2}+\\left(\\int_t^T b^2_s\\ \\mathrm{d}s\\right)^{q\/2}\\right]$$\nfor each $a_s,b_s\\in L^2([t,T])$, by a similar argument to that in case $(i)$ we can deduce that for each $n\\geq 1$ and $t\\in [\\bar T_1,T]$,\n\\begin{equation}\n\\Dis\\sup_{n\\geq 2}\\sup_{i\\geq 1}\\left(\\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\hat y^{n,i}_r|^q\\right]+\\E\\left[\\left(\\int_t^T | \\hat z^{n,i}_s|^2\\ {\\rm\nd}s\\right)^{q\/2}\\right]\\right)\\\\\n\\leq \\Dis 12AT+\\E\\left[\\left(\\int_0^T |z^2_s- z^1_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]<+\\infty.\n\\end{equation}\n\nNow, in view of (53), by first taking supremum with respect to $i$ and then taking $\\limsup$ with respect to $n$ in (51) and finally using Fatou's lemma, the monotonicity and continuity of the function $\\kappa(\\cdot)$ together with Bihari's inequality, we can deduce the existence of processes $(Y_t,Z_t)_{t\\in [\\bar T_1,T]}\\in {\\s}^q(\\bar T_1,T;\\R^k)\\times {\\rm M}^q(\\bar T_1,T;\\R^{k\\times d})$ such that\n\\begin{equation}\n\\Lim \\E\\left[\\sup_{t\\in [\\bar T_1,T]}|(y^n_t-y^1_t)-Y_t|^q+\\left(\\int_{\\bar T_1}^T\n|(z^n_t-z^1_t)-Z_t|^2\\mathrm{d}t\\right)^{q\/2}\\right]=0.\n\\end{equation}\nThus, note that $(y^1_t,z^1_t)_{t\\in\\T}\\in \\s^\\beta\\times \\M^\\beta$ for each $\\beta\\in (0,1)$ and $(y^1_t)_{t\\in\\T}$ belongs to the class (D). By passing to the limit in ucp for BSDE (34), in view of (54), (H2), (H3) and Lebesgue's dominated convergence theorem, we deduce that\n$\n(y_t,z_t)_{t\\in [\\bar T_1,T]}:=(Y_t+y^1_t,Z_t+z^1_t)_{t\\in [\\bar T_1,T]}\n$\nis an $L^1$ solution to the\nBSDE with parameters $(\\xi,T,g)$ on $[\\bar T_1,T]$.\\vspace{0.1cm}\n\nFinally, noticing that the positive real number $\\overline{\\delta T}$ depends only on $q$, $\\lam$ and $A$, we can find a minimal integer $\\bar N\\geq 1$ such that $T_{\\bar N}$=0. Thus, we can repeat, in finite steps, the above procedure to obtain an $L^1$ solution to BSDE (1) on $[\\bar T_2,\\bar T_1]$, $[\\bar T_3,\\bar T_2]$, $\\cdots$, $[0,\\bar T_{\\bar N-1}]$, and then we find an $L^1$ solution to BSDE (1) on $[0,T]$. The proof of Theorem 2 is finally completed.\\vspace{0.2cm} \\hfill $\\Box$\n\n{\\bf Remark 4}\\ \\ We would like to mention that it is interesting that in the case of $\\alpha$ in (H3) values in $[1\/2,1)$, the assumption (H1) in Theorem 2 needs to be replaced with the stronger assumption (H1)${}_{\\bar p}$. The main reason is to ensure obtaining the key inequality (49). This indicates the difference between Propositions 3 and 4. In addition, we point out that how to divide appropriately the time interval $\\T$ is also one of key problems in the proof of Theorem 2.\n\n\n\\section{Existence and Uniqueness of the solution in $\\s^1\\times{\\rm M^1}$ and examples}\n\nIn this section, by virtue of Theorems 1 and 2 we will establish an existence and uniqueness result of the solution in the space $\\s^1\\times {\\rm M}^1$ (a new type of $L^1$ solution) for multidimensional BSDEs with generators of one-sided Osgood type. This is the first time to the best of our knowledge. We will also provide two examples in this section to illustrate our theoretical results.\\vspace{0.2cm}\n\n{\\bf Theorem 3} Assume that the generator $g$ satisfies assumptions (H1)-(H3). In the case when the $\\alpha$ defined in (H3) values in $[1\/2,1)$, we also assume that there exists a constant $\\bar p>1$ such that the function $\\rho(\\cdot)$ in (H1) satisfies (14). If the following assumption (H4) holds true:\\\\\n\n{\\bf (H4)}\\ $\\Dis \\E\\left[\\sup\\limits_{t\\in\\T}\\left(\\E\\left[|\\xi|\n+\\left.\\int_0^T |g(s,0,0)|\\ {\\rm d}s\\right|\\F_t\\right]\\right)\\right]<+\\infty.\n\\vspace{0.4cm}\n$\n\n\\noindent then BSDE (1) admits a unique solution $(y_t,z_t)_{t\\in\\T}$ in $\\s^1(0,T;\\R^k)\\times{\\rm M}^1(0,T;\\R^{k\\times d})$.\\vspace{0.3cm}\n\n{\\bf Proof}\\ \\ It is clear that the uniqueness follows from Theorem 1 directly. Note that if (H4) holds true, then $\\xi\\in \\LT$. It follows from Theorems 1-2 that BSDE (1) admits a unique $L^1$ solution $(y_t,z_t)_{t\\in\\T}$, i.e., $(y_t)_{t\\in\\T}$ belongs to the class (D) and $(y_t,z_t)_{t\\in\\T}\\in \\s^\\beta\\times {\\rm M}^\\beta$ for each $\\beta\\in (0,1)$. Hence, in order to complete the proof of Theorem 3, it remains to show, under (H1)-(H4),\n$$(y_t,z_t)_{t\\in\\T}\\in \\s^1(0,T;\\R^k)\\times{\\rm M}^1(0,T;\\R^{k\\times d}).$$\nIn fact, let us fix $n\\geq 1$ and denote $\\tau_n$ the stopping time\n$$\n\\tau_n:=\\inf\\left\\{t\\in\\T:\\int^t_{0}|z_s|^2\\\n\\mathrm{d}s\\geq n\\right\\}\\wedge T.\n$$\nBy Corollary 2.3 in \\cite{Bri03} we know that for each $t\\in\\T$,\\vspace{0.1cm}\n\\begin{equation}\n|y_{t\\wedge \\tau_n}|\\leq |y_{\\tau_n}|+\\int_{t\\wedge \\tau_n}^{\\tau_n}\\left\\langle {y_s\\over |y_s|}\\mathbbm{1}_{|y_s|\\neq 0}, g(s,y_s,z_s)\\right\\rangle{\\rm d}s-\\int_{t\\wedge \\tau_n}^{\\tau_n}\\left\\langle {y_s\\over |y_s|}\\mathbbm{1}_{|y_s|\\neq 0},z_s {\\rm d}B_s\\right\\rangle.\\vspace{0.1cm}\n\\end{equation}\nAnd, it follows from (H1) and (H3) that $\\ass$,\n\\begin{equation}\n\\left\\langle {y_s\\over |y_s|}\\mathbbm{1}_{|y_s|\\neq 0}, g(s,y_s,z_s)\\right\\rangle\\leq \\rho(|y_s|)+|g(s,0,0)|+\\gamma(g_s+|z_s|)^{\\alpha}.\n\\end{equation}\nThus, in view of (55) and (56), using a similar argument to that in the proof of Theorem 1 we can get that for each $t\\in \\T$,\n\\begin{equation}\n|y_t|\\leq (AT+\\bar G(t))\\cdot e^{AT},\n\\end{equation}\nwhere\n$$\\bar G(t):=\\E\\left[|\\xi|+\\left.\\int_0^T|g(s,0,0)|\\ \\mathrm{d}s+\\gamma\\int_0^T(g_s+|z_s|)^{\\alpha}\\\n\\mathrm{d}s\\right|\\F_t\\right],\\ \\ t\\in \\T.\\vspace{0.2cm}\n$$\nFurthermore, it follows from (H4) and a similar argument to obtain (5) that $\\bar G(\\cdot)\\in \\s^1(0,T;\\R^k)$ and then, in view of (57), $(y_t)_{t\\in\\T}\\in \\s^1(0,T;\\R^k)$. Finally, note by (H1) and (H3) together with Remark 1 that $\\as$, for each $(y,z)\\in \\R^k\\times\\R^{k\\times d}$,\n$$\n\\langle y, g(t,y,z) \\rangle\\leq \\bar\\kappa(|y|^2)+\\lambda |y||z|+|y||g(t,0,0)|\\leq A|y|^2+\\lambda |y||z|+|y||g(t,0,0)|+A,\n$$\nwhere the function $\\bar\\kappa(\\cdot)$ is defined in (H1a)$_2$. It follows from Proposition 1 with $p=1$ and $u=t=0$ that $(z_t)_{t\\in\\T}\\in {\\rm M}^1(0,T;\\R^{k\\times d})$. Theorem 3 is then proved.\\vspace{0.2cm} \\hfill $\\Box$\n\nBy Theorems 1-3 and Remarks 1 and 2 the following corollary is immediate.\\vspace{0.1cm}\n\n{\\bf Corollary 1} Assume that the generator $g$ satisfies assumption (H1b)$_p$ for some $p>1$, (H2) and (H3). Then, for each $\\xi\\in\\LT$, BSDE (1) admits a unique $L^1$ solution $(y_t,z_t)_{t\\in\\T}$. Furthermore, if (H4) holds true, then\n$(y_t,z_t)_{t\\in\\T}\\in \\s^1(0,T;\\R^k)\\times {\\rm M}^1(0,T;\\R^{k\\times d})$.\\vspace{0.2cm}\n\n{\\bf Remark 5} Note that if the generator $g$ satisfies the monotonicity condition used in \\citet{Bri03}, then it must satisfy (H1b)$_p$ for all $p>1$. Theorems 1-3 of this paper generalize Theorems 6.2 and 6.3 in \\cite{Bri03}.\\vspace{0.2cm}\n\n{\\bf Example 1} Let $k=1$ and\n$$\ng(\\omega,t,y,z)=h(|y|)-e^{|B_t(\\omega)|\\cdot y}+(e^{-y}\\wedge 1)\\cdot \\sin |z|+{1\\over \\sqrt{t}}\\mathbbm{1}_{t>0},\n$$\nwhere\n$$\nh(x)=\\left\\{\n\\begin{array}{lll}\n-x|\\ln x|& ,&0<x\\leq \\delta;\\\\\nh'(\\delta-0)(x-\\delta)+h(\\delta)& ,&x> \\delta;\\\\\n0& ,&{\\rm other\\ cases}\n\\end{array}\\right.\\vspace{0.1cm}\n$$\nwith $\\delta>0$ small enough.\n\nIt is not hard to check that $g$ satisfies assumptions (H2) and (H3) with $\\lam =1$ and any $\\alpha\\in (0, 1\/2)$. Furthermore, note that $e^{-\\beta y}$ is decreasing in $y$ for each $\\beta\\geq 0$, $h(\\cdot)$ is concave and sub-additive and then the following inequality holds: $\\as$,\n$$\n\\RE\\ y_1,y_2,z,\\ \\\n\\left\\langle {y_1-y_2\\over |y_1-y_2|}\\mathbbm{1}_{|y_1-y_2|\\neq 0},\ng(\\omega,t,y_1,z)-g(\\omega,t,y_2,z)\\right\\rangle\\leq\nh(|y_1-y_2|)\n$$\nwith $\\int_{0^+} {{\\rm d}u\\over h(u)}=+\\infty$.\nIt follows that $g$ also satisfies assumption (H1). Then, by Theorems 1-2 we know that for each $\\xi\\in\\LT$, the BSDE with the parameters $(\\xi,T,g)$ admits a unique $L^1$ solution $(y_t,z_t)_{t\\in\\T}$. Moreover, by Theorem 3 we also know that if (H4) holds true for $\\xi$ and $g(t,0,0)$, then $$(y_t,z_t)_{t\\in\\T}\\in \\s^1(0,T;\\R^k)\\times {\\rm M}^1(0,T;\\R^{k\\times d}).$$\n\n{\\bf Example 2}\\ Let $y=(y_1,\\cdots,y_k)$ and\n$g(\\omega,t,y,z)=(g_1(\\omega,t,y,z),\\cdots,g_k(\\omega,t,y,z))$, where for each $i=1,\\cdots,k$,\n$$\ng_i(\\omega,t,y,z):=e^{-y_i}+\\bar h(|y|)+\\left(|z|^2\\wedge |z|^{2\/3}\\right)+|B_t(\\omega)|,\n$$\nwith\n$$\n\\bar h(x)=\\left\\{\n\\begin{array}{lll}\n-x|\\ln x|^{1\/p}& ,&0<x\\leq \\delta;\\\\\n\\bar h'(\\delta-0)(x-\\delta)+\\bar h(\\delta)& ,&x> \\delta;\\\\\n0& ,&{\\rm other\\ cases}\n\\end{array}\\right.\\vspace{0.2cm}\n$$\nwith $\\delta>0$ small enough and $p>1$.\n\nIn the same way as in Example 1, we can check that this generator $g$ satisfies assumptions (H1b)$_p$ with function $\\bar h(\\cdot)$, (H2) and (H3) with $\\lam =1$ and $\\alpha=2\/3$. It then follows from Corollary 1 that for each $\\xi\\in\\LT$, the BSDE with the parameters $(\\xi,T,g)$ admits a unique $L^1$ solution $(y_t,z_t)_{t\\in\\T}$. And, if (H4) holds true for $\\xi$ and $g(t,0,0)$, then $(y_t,z_t)_{t\\in\\T}\\in \\s^1(0,T;\\R^k)\\times {\\rm M}^1(0,T;\\R^{k\\times d}).$\n\n\\section{Stability of the $L^1$ solutions and the solutions in $\\s^1\\times {\\rm M}^1$}\n\nIn this section, enlightened by the proof of Proposition 5 and Theorems 2-3, we shall put forward and prove the stability theorems of the $L^1$ solutions and the solutions in the space $\\s^1\\times {\\rm M}^1$ for multidimensional BSDEs with generators of one-sided Osgood type. To the best of our knowledge, this is the first time for the $L^1$ solution of multidimensional BSDEs.\\vspace{0.2cm}\n\nIn the sequel, for each $m\\in \\N$, let $\\xi^m\\in \\LT$ and let  $(y_t^m, z_t^m)_{t\\in [0,T]}$ be\nan $L^1$ solution of\nthe following BSDEs depending on parameter $m$:\n\\begin{equation}\ny_t^m=\\xi^m+\\int_t^T g^m(s,y_s^m,z_s^m)\\ {\\rm d}s -\\int_t^T z_s^m{\\rm d}B_s,\\quad t\\in [0,T].\n\\end{equation}\nFurthermore,  we introduce  the following assumptions:\\vspace{0.2cm}\n\n{\\bf (B1)}\\ All $g^m$ satisfy assumptions (H1)-(H3)\n with the same parameters $\\rho(\\cdot)$, $\\lambda$, $\\gamma$, $g_t$ and $\\alpha$. Furthermore, in the case of $\\alpha\\in [1\/2,1)$ we assume that $g$ satisfies (H1) with a function $\\rho(\\cdot)$ satisfying (14) for some $\\bar p>1$. \\vspace{0.2cm}\n\n{\\bf (B2)}\\ There exists a nonnegative real number sequence\n$\\{a_m\\}_{m=1}^{+\\infty}$ satisfying $\\lim\\limits_{m\\To \\infty}a_m=0$ such that\n$\\as$,  for each $m\\geq 1$,\n\\begin{equation}\n\\RE\\ (y,z)\\in \\R^k\\times \\R^{k\\times d},\\ \\ |g^m(\\omega,t,y,z)-g^0(\\omega,t,y,z)|\\leq a_m.\n\\end{equation}\nAnd,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[|\\xi^{m}-\\xi^{0}|\\right]=0.\n\\vspace{0.3cm}\n\\end{equation}\n\nThe following Theorem 4 is the stability theorem of $L^1$ solutions.\\vspace{0.1cm}\n\n{\\bf Theorem 4}\\ Under assumptions (B1) and (B2), we have\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\sup\\limits_{t\\in [0,T]}\\E\\left[|y_t^m-y_t^0|\\right]=0,\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^m-y_t^0|^\\beta+\\left(\\int_0^T |z_s^m -z_s^0|^2 {\\rm d}s\\right)^{\\beta\/2} \\right]=0.\\vspace{0.1cm}\n\\end{equation}\n\n{\\bf Proof}\\ \\ For each $ m\\in \\N$, set $(y^{m,0}_\\cdot,z^{m,0}_\\cdot):=(0,0)$ and, similar to the beginning part of the proof of Theorem 2, define recursively the process sequence $\\{(y^{m,n}_\\cdot,z^{m,n}_\\cdot)\\}_{n=1}^{+\\infty}$ by the $L^1$ solutions of the following BSDEs\n\\begin{equation}\ny_t^{m,n}=\\xi^m+\\int_t^T g^{m,n}(s,y_s^{m,n},z_s^{m,n-1})\\ {\\rm d}s -\\int_t^T z_s^{m,n}{\\rm d}B_s,\\ \\  t\\in [0,T],\\vspace{0.2cm}\n\\end{equation}\nwhere $(y^{m,n}_t,z^{m,n}_t)_{t\\in [0,T]}\\in \\s^\\beta\\times \\M^\\beta$ for each $\\beta\\in (0,1)$ and $(y^{m,n}_t)_{t\\in\\T}$ belongs to the class (D) for each $m,n\\in \\N$.\\vspace{0.2cm}\n\nIn the sequel, note that $\\as$,\n\\begin{equation}\n|y_t^m -y_t^0| \\leq |y_t^m -y_t^{m,n}| + |y_t^{m,n} -y_t^{0,n}|+|y_t^{0,n} -y_t^0|\\leq \\Dis 2\\sup_{m\\geq 0}|y_t^{m,n}-y_t^m|+|y_t^{m,n} -y_t^{0,n}|\n\\end{equation}\nand\n\\begin{equation}\n|z_t^m -z_t^0|\\leq |z_t^m -z_t^{m,n}| + |z_t^{m,n} -z_t^{0,n}|+|z_t^{0,n} -z_t^0|\\leq \\Dis 2\\sup_{m\\geq 0}|z_t^{m,n}-z_t^m|+|z_t^{m,n} -z_t^{0,n}|.\n\\end{equation}\nWe will estimate, respectively, every term of the right hand in (64) and (65).\\vspace{0.2cm}\n\nFirstly, the following Proposition 6 gives the  estimates with respect to the second term of the right hand in (64) and (65).\\vspace{0.1cm}\n\n{\\bf Proposition 6}\\ \\ For each $n\\geq 1$, we have\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\sup\\limits_{t\\in [0,T]}\\E\\left[|y_t^{m,n}-y_t^{0,n}|\\right]=0,\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^{m,n}-y_t^{0,n}|^\\beta+\\left(\\int_0^T |z_s^{m,n}-z_s^{0,n}|^2\\  {\\rm d}s\\right)^{\\beta\/2} \\right]=0.\n\\end{equation}\n\n{\\bf Proof}\\ \\ We first consider the case of $n=1$. Let us fix $k,m\\geq 1$ and denote $\\tau^m_k$ the stopping time\n$$\n\\tau^m_k:=\\inf\\left\\{t\\in\\T:\\int^t_{0}(|z^{m,1}_s|^2\n+|z^{0,1}_s|^2)\\ \\mathrm{d}s\\geq k\\right\\}\\wedge T.\n$$\nCorollary 2.3 in \\cite{Bri03} leads to the following inequality\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis|\\hat{y}^{m,1}_{t\\wedge\\tau^m_k}|&\\leq &\\Dis |\\hat{y}^{m,1}_{\\tau^m_k}|+\\int^{\\tau^m_k}_{t\\wedge\\tau^m_k}\n\\left\\langle {\\hat{y}^{m,1}_s\\over |\\hat{y}^{m,1}_s| }\\mathbbm{1}_{|\\hat{y}^{m,1}_s|\\neq 0},g^m(s,y^{m,1}_s,0)-g^0(s,y^{0,1}_s,0)\\right\\rangle\\\n\\mathrm{d}s\\vspace{0.1cm}\\\\\n&&\\Dis -\\int^{\\tau^m_k}_{t\\wedge\\tau^m_k}\\left\\langle {\\hat{y}^{m,1}_s\\over |\\hat{y}^{m,1}_s|}\\mathbbm{1}_{|\\hat{y}^{m,1}_s|\\neq 0},\\hat{z}^{m,1}_s\\ \\mathrm{d} B_s\\right\\rangle,\\ \\ t\\in\\T,\n\\end{array}\n\\end{equation}\nwhere and hereafter\n$$\\hat{y}^{m,1}_\\cdot:=y^{m,1}_\\cdot-y^{0,1}_\\cdot\\ \\ {\\rm and}\\ \\ \\hat{z}^{m,1}_\\cdot:=z^{m,1}_\\cdot-z^{0,1}_\\cdot.$$\nAnd, it follows from assumption (H1) of $g^m$  and (59) that $\\ass$,\n\\begin{equation}\n\\begin{array}{ll}\n&\\Dis \\left\\langle{\\hat{y}^{m,1}_s\\over |\\hat{y}^{m,1}_s|} \\mathbbm{1}_{|\\hat{y}^{m,1}_s|\\neq 0},g^m(s,y^{m,1}_s,0)-g^0(s,y^{0,1}_s,0)\n\\right\\rangle\\vspace{0.1cm}\\\\\n\\leq &\\Dis \\left\\langle{\\hat{y}^{m,1}_s\\over |\\hat{y}^{m,1}_s|} \\mathbbm{1}_{|\\hat{y}^{m,1}_s|\\neq 0},g^m(s,y^{m,1}_s,0)-g^m(s,y^{0,1}_s,0)\n\\right\\rangle+|g^m(s,y^{0,1}_s,0)-g^0(s,y^{0,1}_s,0)|\n\\vspace{0.1cm}\\\\\n\\leq &\\Dis \\rho(|\\hat{y}^{m,1}_s|)+a_m.\n\\end{array}\n\\end{equation}\nThen, combining (68) with (69) we can deduce that for each $k,m\\geq 1$,\n\\begin{equation}\n|\\hat{y}^{m,1}_{t\\wedge\\tau^m_k}|\\leq a_mT+\\E\\left[\n|\\hat{y}^{m,1}_{\\tau^m_k}|\n+\\left.\\int^{\\tau^m_k}_{t\\wedge\\tau^m_k}\n\\rho(|\\hat{y}^{m,1}_s|)\\ \\mathrm{d}s\\right|\\F_t\\right],\\ \\ t\\in\\T.\n\\end{equation}\nSince $\\hat y^{m,1}_\\cdot$ belongs to the class (D), and $\\rho(\\cdot)$ increases at most linearly, we can send $k$ to $+\\infty$ in (70) and use Lebesgue's dominated convergence theorem, in view of $\\tau^m_k\\To T$ as $k\\To \\infty$ and $\\hat y^{m,1}_T=\\xi^m-\\xi^0$, to get that for each $m\\geq 1$,\n\\begin{equation}\n|\\hat{y}^{m,1}_t| \\leq H_m(t) +\\E\\left[\\left.\\int_t^T\\rho(|\\hat{y}^{m,1}_s|)\\ \\mathrm{d}s\\right|\\F_t\\right],\\ \\ t\\in \\T,\n\\end{equation}\nwhere\n$$H_m(t):=a_mT+\\E\\left[\\left.|\\xi^m-\\xi^0|\n\\right|\\F_t\\right].\\vspace{0.2cm}$$\nIn the sequel, note by Lemma 6.1 in \\cite{Bri03} and assumption (B2) that\n\\begin{equation}\n\\RE\\ \\beta\\in (0,1),\\ \\ \\sup_{m\\geq 1}\\E\\left[\\sup_{t\\in\\T}|H_m(t)|^\\beta\n\\right]\\leq {1\\over 1-\\beta}\\sup_{m\\geq 1} \\left(\\E\\left[H_m(T)\\right]\\right)^\\beta<+\\infty\n\\end{equation}\nand\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty} \\E\\left[|H_m(T)|\\right]=0.\\vspace{0.2cm}\n\\end{equation}\nArguing as that from (18) to (29) we can deduce that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\left(\\sup\\limits_{t\\in [0,T]}\\E\\left[|\\hat y_t^{m,1}|\\right]+\n\\E\\left[\\sup\\limits_{t\\in [0,T]} |\\hat y_t^{m,1}|^\\beta\\right]\\right)=0.\n\\end{equation}\nFurthermore, note that for each $m\\geq 1$, $(\\hat y_t^{m,1},\\hat z_t^{m,1})_{t\\in\\T}$ is an $L^1$ solution of the following BSDE:\n\\begin{equation}\n\\hat{y}_t^{m,1}=\\xi^{m}-\\xi^0+\\int^{T}_{t}\\hat g^{m,1}(s,\\hat y_s^{m,1})\\ \\mathrm{d} s-\\int^{T}_{t}\\hat{z}_s^{m,1}\\mathrm{d}B_s, \\quad t\\in\\T,\n\\end{equation}\nwhere for each $y\\in\\R^k$,\n$\n\\hat g^{m,1}(t,y):=g^{m}(t,y+y^{0,1}_t,0)-g^0(t,y^{0,1}_t,0).\n$\nIt follows from assumption (H1) on $g^m$ together with (59) that $\\as$, for each $m\\geq 1$ and $y\\in \\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\hat g^{m,1}(t,y)\\rangle&\\leq &\\Dis \\langle y,g^{m}(t,y+y^{0,1}_t,0)-g^{m}(t,y^{0,1}_t,0)\\rangle +|y||g^{m}(t,y^{0,1}_t,0)-g^0(t,y^{0,1}_t,0)|\\\\\n&\\leq &\\Dis \\kappa(|y|^2)+|y|a_m,\n\\end{array}\n\\end{equation}\nwhere the function $\\kappa(\\cdot)$ is defined in (H1a)$_2$. Thus, in view of (74)-(76), Proposition 1, (B2) and the assumption of $\\kappa(\\cdot)$, a similar argument to that from (30) to (33) yields that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty} \\E\\left[\\left(\\int_0^T |\\hat z^{m,1}_s|^2\\\n{\\rm d}s\\right)^{\\beta\/2}\\right]=0.\n\\end{equation}\nFrom (74) and (77), we know that (66) and (67) hold true for $n=1$.\\vspace{0.2cm}\n\nNow, let us fix arbitrarily a $n\\geq 2$ and assume that (66) and (67) hold true for $n-1$. In the sequel, we will prove that they also hold for $n$. For each $m\\geq 1$, define\n$$\\hat{y}^{m,n}_\\cdot:=y^{m,n}_\\cdot-y^{0,n}_\\cdot\\ \\ {\\rm and}\\ \\ \\hat{z}^{m,n}_\\cdot:=z^{m,n}_\\cdot-z^{0,n}_\\cdot.$$\nThen, $(\\hat y_t^{m,n},\\hat z_t^{m,n})_{t\\in\\T}$ is an $L^1$ solution of the following BSDE:\n\\begin{equation}\n\\hat{y}_t^{m,n}=\\xi^{m}-\\xi^0+\\int^{T}_{t}\\hat g^{m,n}(s,\\hat y_s^{m,n})\\ \\mathrm{d} s-\\int^{T}_{t}\\hat{z}_s^{m,n}\\ \\mathrm{d}B_s, \\quad t\\in\\T,\n\\end{equation}\nwhere for each $y\\in\\R^k$,\n$\n\\hat g^{m,n}(t,y):=g^{m}(t,y+y^{0,n}_t,z^{m,n-1}_t)\n-g^0(t,y^{0,n}_t,z^{0,n-1}_t).\n$\nIt follows from assumption (H1) of $g^m$ together with Remark 1 and (59) that $\\as$, for each $m\\geq 1$ and $y\\in \\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\hat g^{m,n}(t,y)\\rangle&\\leq &\\Dis \\langle y,g^{m}(t,y+y^{0,n}_t,z^{m,n-1}_t)-\ng^{m}(t,y^{0,n}_t,z^{m,n-1}_t)\\rangle\\vspace{0.1cm}\\\\\n&&\\Dis +|y||g^{m}(t,y^{0,n}_t,z^{m,n-1}_t)\n-g^m(t,y^{0,n}_t,z^{0,n-1}_t)|\\vspace{0.1cm}\\\\\n&&\\Dis +|y||g^{m}(t,y^{0,n}_t,z^{0,n-1}_t)\n-g^0(t,y^{0,n}_t,z^{0,n-1}_t)|\\vspace{0.1cm}\\\\\n&\\leq &\\Dis \\kappa(|y|^2)+|y|\\cdot(\\triangle_t^{m,n}+a_m),\n\\end{array}\n\\end{equation}\nwhere the function $\\kappa(\\cdot)$ is defined in (H1a)$_2$, and, in view of (H3),\n\\begin{equation}\n\\triangle_t^{m,n}:=|g^{m}(t,y^{0,n}_t,z^{m,n-1}_t)\n-g^m(t,y^{0,n}_t,z^{0,n-1}_t)|\\leq \\lambda |z^{m,n-1}_t-z^{0,n-1}_t|\n\\end{equation}\nas well as\n\\begin{equation}\n\\triangle_t^{m,n}\\leq 2\\gamma (g_t+|y^{0,n}_t|+|z^{m,n-1}_t|+|z^{0,n-1}_t|)^\\alpha.\n\\end{equation}\nSince (67) holds true for $n-1$, from (80) we know that the sequence of random variables $\\{\\int_0^T \\triangle_t^{m,n}\\ {\\rm d}t\\}_{m=1}^{+\\infty}$ converges in probability to $0$ as $m\\To \\infty$, and from (81) and H\\\"{o}lder's inequality that for each $q>1$ satisfying $\\alpha q<1$,\n\\begin{equation}\n\\sup_{m\\geq 1}\\E\\left[\\left(\\int_0^T \\triangle_t^{m,n}\\ {\\rm d}t\\right)^q\\right]\\leq K_n\\left(1+\\sup_{m\\geq 1}\\E\\left[\\left(\\int_0^T|z^{m,n-1}_t|^2 \\ {\\rm d}t\\right)^{\\alpha q\\over 2}\\right] \\right)<+\\infty,\n\\end{equation}\nwhere $K_n>0$ is a constant independent of $m$. Hence, for each $q'>1$ satisfying $\\alpha q'<1$, we have\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty} \\E\\left[\\left(\\int_0^T\\triangle_t^{m,n} \\\n{\\rm d}t\\right)^{q'}\\right]=0.\\vspace{0.1cm}\n\\end{equation}\nOn the other hand, note from assumption (H1) of $g^m$  and (79) that $\\as$,\n\\begin{equation}\n\\left\\langle{\\hat{y}^{m,n}_t\\over |\\hat{y}^{m,n}_t|} \\mathbbm{1}_{|\\hat{y}^{m,n}_t|\\neq 0},g^m(t,y^{m,n}_t,z^{m,n-1}_t)\n-g^0(t,y^{0,n}_t,z^{0,n-1}_t)\n\\right\\rangle\\leq \\rho(|\\hat{y}^{m,n}_t|)+\\triangle_t^{m,n}+a_m.\n\\end{equation}\nArguing as that from (68) to (71), in view of (82), (83) and (84), we can obtain that for each $m\\geq 1$,\n\\begin{equation}\n|\\hat{y}^{m,n}_t| \\leq H^n_m(t) +\\E\\left[\\left.\\int_t^T\\rho(|\\hat{y}^{m,n}_s|)\\ \\mathrm{d}s\\right|\\F_t\\right],\\ \\ t\\in \\T,\n\\end{equation}\nwhere\n$$H^n_m(t):=a_mT+\\E\\left[\\left.|\\xi^m-\\xi^0|\n+\\int_0^T\\triangle_s^{m,n} \\\n{\\rm d}s\\right|\\F_t\\right].\\vspace{0.1cm}$$\nFurthermore, in view of (72), (73), (82), (83) and (85), a similar argument to that from (18) to (29) yields that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\left(\\sup\\limits_{t\\in [0,T]}\\E\\left[|\\hat y_t^{m,n}|\\right]+\n\\E\\left[\\sup\\limits_{t\\in [0,T]} |\\hat y_t^{m,n}|^\\beta\\right]\\right)=0.\n\\end{equation}\nFinally, in view of (78), (79), (83), Proposition 1, (B2) and the assumption of $\\kappa(\\cdot)$, using a similar argument to that from (30) to (33) yields that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty} \\E\\left[\\left(\\int_0^T |\\hat z^{m,n}_s|^2\\\n{\\rm d}s\\right)^{\\beta\/2}\\right]=0.\n\\end{equation}\nIn view of (86) and (87), we have proved that (66) and (67) hold also true for $n$. Thus, by induction the proof of Proposition 6 is completed.\\hfill $\\Box$\\vspace{0.2cm}\n\nNext, we turn to the estimates with respect to the first term of the right hand in (64) and (65).\\vspace{0.2cm}\n\n{\\bf Proposition 7}\\ \\ In the case when the $\\alpha$ defined in (H3) values in $(0,1\/2)$, there exists a positive real number $\\delta T>0$ depending only on $\\lam$ and $A$ such that for each $t\\in [T_1, T]$ with $T_1:=(T-\\delta T)\\vee 0$,\n\\begin{equation}\n\\Lim \\sup_{m\\geq 0}\\E\\left[\\sup_{r\\in [t,T]}|y_r^{m,n}-y_r^m|^2+\\int_t^T |z_s^{m,n}-z_s^m|^2\\ {\\rm d}s\\right]=0.\n\\end{equation}\nIn the case when the $\\alpha$ defined in (H3) values in $[1\/2,1)$, for each $q\\in (1,\\bar p\\wedge {1\\over \\alpha})$, there exists a positive real number $\\overline{\\delta T}>0$ depending only on $q$, $\\lam$ and $A$ such that for each $t\\in [\\bar T_1, T]$ with $\\bar T_1:=(T-\\overline{\\delta T})\\vee 0$,\n\\begin{equation}\n\\Lim \\sup_{m\\geq 0}\\E\\left[\\sup_{r\\in [t,T]}|y_r^{m,n}-y_r^m|^q+\\left(\\int_t^T |z_s^{m,n}-z_s^m|^2\\ {\\rm d}s\\right)^{q\/2}\\right]=0.\\vspace{0.2cm}\n\\end{equation}\n\n{\\bf Proof}\\ \\ We only prove the case when the $\\alpha$ in (H3) values in $(0,1\/2)$. In view of the proof of case $(ii)$ in Theorem 2, another case can be proved in the same way.\\vspace{0.2cm}\n\nNow, we assume that $\\alpha\\in (0,1\/2)$ and set $\\bar{y}^{m,n}_\\cdot:=y^{m,n}_\\cdot-y^m_\\cdot$ and $\\bar z^{m,n}_\\cdot:=z^{m,n}_\\cdot-z^m_\\cdot$ for each $m,n\\geq 0$. Note that for each $m\\geq 0$ and $n\\geq 1$, $(\\bar y_t^{m,n},\\bar z_t^{m,n})_{t\\in\\T}$ is a solution of the following BSDE:\n\\begin{equation}\n\\bar{y}_t^{m,n}=\\int_t^T \\bar{g}^{m,n}(s,\\bar{y}_s^{m,n})\\ {\\rm\nd}s-\\int_t^T \\bar{z}_s^{m,n}{\\rm d}B_s,\\ \\ \\ t\\in \\T,\n\\end{equation}\nwhere for each $y\\in \\R^k$,\n$\\bar{g}^{m,n}(s,y):=g^m(s,y+y_s^m,z^{m,n-1}_s)-\ng^m(s,y_s^m,z^m_s).$\nIt follows from (H1) of $g^m$ and Remark 1 that $\\as$, for each $y\\in\\R^k$,\\vspace{-0.2cm}\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\bar{g}^{m,n}(t,y)\\rangle &\\leq &\\langle y,g^m(t,y+y_t^m,z^{m,n-1}_t)-g^m(t,y_t^m,z^{m,n-1}_t)\n\\rangle\\\\\n&& \\Dis +|y||g^m(t,y_t^m,z^{m,n-1}_t)-g^m(t,y_t^m,z^m_t)|\\\\\n&\\leq & \\Dis \\kappa(|y|^2)+|y|\\bar\\triangle^{m,n}_t\n\\end{array}\n\\end{equation}\nwhere $\\kappa(\\cdot)$ is defined in (H1a)$_2$, and in view of (H3) of $g^m$,\n\\begin{equation}\n\\bar\\triangle^{m,n}_t:=g^m(t,y_t^m,z^{m,n-1}_t)-g^m(t,y_t^m,z^m_t)\\leq \\lam |z^{m,n-1}_t-z^m_t|=\\lam |\\bar z^{m,n-1}_t|\n\\end{equation}\nas well as\n\\begin{equation}\n\\bar\\triangle^{m,n}_t\\leq 2\\gamma \\left(g_t+\n|y^m_t|+|z^m_t|+|z^{m,n-1}_t|\\right)^\\alpha.\\vspace{0.2cm}\n\\end{equation}\nIn view of $\\alpha\\in (0,1\/2)$ and (93), using H\\\"{o}lder's inequality, Jensen's inequality and Doob's inequality yields that for each $m\\geq 0$ and $n\\geq 1$,\n\\begin{equation}\n\\E\\left[\\left(\\int_0^T \\bar\\triangle^{m,n}_t\\ {\\rm d}t\\right)^2\\right]<+\\infty\\ \\ {\\rm and\\ then}\\ \\\n\\E\\left[\\left.\\int_0^T \\bar\\triangle^{m,n}_s\\ {\\rm d}s\\right|\\F_t\\right]\\in \\s^2.\n\\end{equation}\nOn the other hand, note from assumption (H1) of $g^m$  and (91) that $\\as$,\n\\begin{equation}\n\\left\\langle{\\bar{y}^{m,n}_t\\over |\\bar{y}^{m,n}_t|} \\mathbbm{1}_{|\\bar{y}^{m,n}_t|\\neq 0},g^m(t,y^{m,n}_t,z^{m,n-1}_t)\n-g^m(t,y^m_t,z^m_t)\n\\right\\rangle\\leq \\rho(|\\bar{y}^{m,n}_t|)+\\bar\\triangle_t^{m,n}.\n\\end{equation}\nThus, in view of (95) and (94), arguing as that from (2) to (7), by virtue of Gronwall's inequality we can obtain that for each $m\\geq 0$ and $n\\geq 1$,\n\\begin{equation}\n(\\bar y_t^{m,n})_{t\\in\\T}\\in \\s^2.\n\\end{equation}\nFurthermore, in view of (96), (91), (94) and Remark 1, it follows from Proposition 1 that $(\\bar z_t^{m,n})_{t\\in\\T}\\in {\\rm M}^2$. Thus, $(\\bar y_t^{m,n},\\bar z_t^{m,n})_{t\\in\\T}$ is an $L^2$ solution of BSDE (90) for each $m\\geq 0$ and $n\\geq 1$.\\vspace{0.1cm}\n\nIn the sequel, in view of (91) and (92), by Proposition 3 and H\\\"{o}lder's inequality we can deduce the existence of a constant $C>0$ such that for each $m\\geq 0$, $n\\geq 2$ and $t\\in [0,T]$,\n\\begin{equation}\n\\hspace*{-0.8cm}\\begin{array}{lll}\n&&\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\bar\ny^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T |\\bar z^{m,n}_s|^2\\ {\\rm d}s\\right]\\vspace{0.1cm}\\\\\n&\\leq &\\Dis e^{C(T-t)}\\left\\{\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\bar y^{m,n}_r|^2 \\right]\\right){\\rm d}s+\\lam^2 (T-t)\\E\\left[\\int_t^T |\\bar z^{m,n-1}_s|^2\\ {\\rm d}s\\right]\\right\\}.\n\\end{array}\n\\end{equation}\nNow, let\n$$\n\\delta T:=\\min\\left\\{{\\ln 2\\over C}, {1\\over 16\\lam^2}, {\\ln 2\\over 2A}\\right\\}\\ \\ {\\rm and}\\ \\\nT_1:=(T-\\delta T)\\vee 0.\\vspace{0.2cm}\n$$\nThen for each $t\\in [T_1,T]$, we have\n\\begin{equation}\ne^{C(T-t)}\\leq 2, \\ \\ \\lam^2e^{C(T-t)}(T-t)\\leq {1\\over 8},\\ \\ e^{2A(T-t)}\\leq 2.\n\\end{equation}\nCombining (97) with (98) yields that for each $m\\geq 0$, $n\\geq 2$ and $t\\in [T_1,T]$,\\vspace{0.2cm}\n\\begin{equation}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\bar\ny^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T |\\bar z^{m,n}_s|^2\\ {\\rm d}s\\right]\n\\leq \\Dis 2\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\bar y^{m,n}_r|^2 \\right]\\right){\\rm d}s+{1\\over 8}\\E\\left[\\int_t^T |\\bar z^{m,n-1}_s|^2\\ {\\rm d}s\\right].\n\\end{equation}\nFurthermore, note by Remark 1 that $\\kappa(x)\\leq A(x+1)$ for each $x\\geq 0$. Gronwall's inequality with (99) and (98) yields that for each $m\\geq 0$, $n\\geq 2$ and $t\\in [T_1,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\bar y^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T | \\bar z^{m,n}_s|^2\\ {\\rm\nd}s\\right]&\\leq &\\Dis \\left(2AT+{1\\over 8}\\E\\left[\\int_t^T |\\bar z^{m,n-1}_s|^2\\ {\\rm d}s\\right]\\right)\\cdot e^{2A(T-t)}\\vspace{0.1cm}\\\\\n&\\leq &\\Dis 4AT+{1\\over 4}\\E\\left[\\int_t^T\n|\\bar z^{m,n-1}_s|^2\\ {\\rm d}s\\right].\n\\end{array}\n\\end{equation}\nBy the above inequality (100) together with the inequality\n$$|\\bar z^{m,n-1}_s|^2\\leq 2(|z^{m,n}_s-z_s^m|^2+|z^{m,n}_s-z_s^{m,n-1}|^2)\n=2(|\\bar z^{m,n}_s|^2+|\\tilde{z}^{m,n}_s|^2),$$\nwe can get that for each $m\\geq 0$, $n\\geq 2$ and $t\\in [T_1,T]$,\n\\begin{equation}\n\\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\bar y^{m,n}_r|^2\\right]+{1\\over 2}\\E\\left[\\int_t^T | \\bar z^{m,n}_s|^2\\ {\\rm d}s\\right]\\leq 4AT+{1\\over 2}\\E\\left[\\int_t^T\n|\\tilde{z}^{m,n}_s|^2\\ {\\rm d}s\\right],\n\\end{equation}\nwhere and hereafter, for each $m\\geq 0$ and $n\\geq 1$ we define\n$$\\tilde{y}^{m,n}_\\cdot:=y^{m,n}_\\cdot-y^{m,n-1}_\\cdot\\ \\ {\\rm and}\\ \\ \\tilde{z}^{m,n}_\\cdot:=z^{m,n}_\\cdot-z^{m,n-1}_\\cdot.\n\\vspace{-0.1cm}$$\n\nNext, as a key step in the proof of Proposition 7 we will show that\n\\begin{equation}\n\\RE\\ t\\in [T_1,T],\\ \\ \\sup_{m\\geq 0}\\sup_{n\\geq 2} \\E\\left[\\int_t^T | \\tilde z^{m,n}_s|^2\\ {\\rm d}s\\right]<+\\infty.\n\\end{equation}\nNote that for each $m\\geq 0$ and $n\\geq 2$, $(\\tilde y_t^{m,n},\\tilde z_t^{m,n})_{t\\in\\T}$ is an $L^1$ solution of the following BSDE:\n\\begin{equation}\n\\tilde{y}_t^{m,n}=\\int_t^T \\tilde{g}^{m,n}(s,\\tilde{y}_s^{m,n})\\ {\\rm\nd}s-\\int_t^T \\tilde{z}_s^{m,n}{\\rm d}B_s,\\ \\ \\ t\\in \\T,\n\\end{equation}\nwhere for each $y\\in \\R^k$,\n$\\tilde{g}^{m,n}(s,y):=g^m(s,y+y_s^{m,n-1},z^{m,n-1}_s)-\ng^m(s,y_s^{m,n-1},z_s^{m,n-2}).$\nIt follows from (H1) of $g^m$ and Remark 1 that $\\as$, for each $y\\in\\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\left\\langle y,\\tilde{g}^{m,n}(t,y)\\right\\rangle &\\leq &\\left\\langle y,g^m(t,y+y_t^{m,n-1},z^{m,n-1}_t)\n-g^m(t,y_t^{m,n-1},z^{m,n-1}_t)\n\\right\\rangle\\vspace{0.1cm}\\\\\n&& \\Dis +|y||g^m(t,y_t^{m,n-1},z^{m,n-1}_t)\n-g^m(t,y_t^{m,n-1},z^{m,n-2}_t)|\\vspace{0.1cm}\\\\\n&\\leq & \\Dis \\kappa(|y|^2)+|y|\\tilde\\triangle^{m,n}_t,\n\\end{array}\n\\end{equation}\nwhere $\\kappa(\\cdot)$ is defined in (H1a)$_2$, and in view of (H3) of $g^m$,\n\\begin{equation}\n\\Dis \\tilde\\triangle^{m,n}_t := \\Dis g^m(t,y_t^{m,n-1},z^{m,n-1}_t)\n-g^m(t,y_t^{m,n-1},z^{m,n-2}_t)\n\\leq \\Dis  \\lam |z^{m,n-1}_t-z^{m,n-2}_t|=\\lam |\\tilde z^{m,n-1}_t|\n\\end{equation}\nas well as\n\\begin{equation}\n\\tilde\\triangle^{m,n}_t\\leq 2\\gamma \\left(g_t+\n|y_t^{m,n-1}|+|z^{m,n-1}_t|+|z^{m,n-2}_t|\\right)^\\alpha.\\vspace{0.2cm}\n\\end{equation}\nIn view of $\\alpha\\in (0,1\/2)$ and (106), using H\\\"{o}lder's inequality, Jensen's inequality and Doob's inequality yields that for each $m\\geq 0$ and $n\\geq 2$,\n\\begin{equation}\n\\E\\left[\\left(\\int_0^T \\tilde\\triangle^{m,n}_t\\ {\\rm d}t\\right)^2\\right]<+\\infty\\ \\ {\\rm and\\ then}\\ \\\n\\E\\left[\\left.\\int_0^T \\tilde\\triangle^{m,n}_s\\ {\\rm d}s\\right|\\F_t\\right]\\in \\s^2.\n\\end{equation}\nOn the other hand, note from assumption (H1) of $g^m$  and (104) that $\\as$,\n\\begin{equation}\n\\left\\langle{\\tilde{y}^{m,n}_t\\over |\\tilde{y}^{m,n}_t|} \\mathbbm{1}_{|\\tilde{y}^{m,n}_t|\\neq 0},g^m(t,y^{m,n}_t,z^{m,n-1}_t)\n-g^m(t,y_t^{m,n-1},z_t^{m,n-2})\n\\right\\rangle\\leq \\rho(|\\tilde{y}^{m,n}_t|)+\\tilde\\triangle_t^{m,n}.\n\\end{equation}\nThus, in view of (108) and (107), arguing as that from (2) to (7), by virtue of Gronwall's inequality we can obtain that for each $m\\geq 0$, $n\\geq 2$ and $t\\in\\T$,\n\\begin{equation}\n|\\tilde y_t^{m,n}|\\leq \\left(AT+\\E\\left[\\left.\\int_0^T \\tilde\\triangle^{m,n}_s\\ {\\rm d}s\\right|\\F_t\\right]\\right)\\cdot e^{AT}\n\\end{equation}\nand then\n\\begin{equation}\n(\\tilde y_t^{m,n})_{t\\in\\T}\\in \\s^2.\n\\end{equation}\nIn view of (110), (104), (107) and Remark 1, it follows from Proposition 1 with $p=2$ and $u=0$ that there exists a constant $C_{A,T}>0$ depending only on $A,T$ such that for each $m\\geq 0$, $n\\geq 2$ and $t\\in \\T$,\n\\begin{equation}\n\\Dis\\E\\left[\\left(\\int_t^T |\\tilde z_s^{m,n}|^2\\ {\\rm d}s\\right)\\right]\\leq \\Dis C_{A,T}\\left\\{\\E\\left[\\sup_{r\\in [t,T]}|\\tilde y_r^{m,n}|^2\\right]+\\E\\left[\\left(\\int_t^T \\tilde \\triangle_s^{m,n}\\ {\\rm d}s\\right)^2\\right]+(AT)^2\\right\\}<+\\infty.\n\\end{equation}\nThus, for each $m\\geq 0$ and $n\\geq 2$, $(\\tilde y_t^{m,n},\\tilde z_t^{m,n})_{t\\in\\T}$ is an $L^2$ solution of BSDE (103). Furthermore, in view of (104) and (105), by Proposition 3 and H\\\"{o}lder's inequality we can deduce that for each $m\\geq 0$, $n\\geq 3$ and $t\\in [0,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n&&\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\tilde\ny^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T |\\tilde z^{m,n}_s|^2\\ {\\rm d}s\\right]\\vspace{0.1cm}\\\\\n&\\leq &\\Dis e^{C(T-t)}\\left\\{\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\tilde y^{m,n}_r|^2 \\right]\\right){\\rm d}s+\\lam^2 (T-t)\\E\\left[\\int_t^T |\\tilde z^{m,n-1}_s|^2\\ {\\rm d}s\\right]\\right\\},\n\\end{array}\n\\end{equation}\nwhere the constant $C>0$ is the same as in (97).\nThen, combining (112) with (98) yields that for each $m\\geq 0$, $n\\geq 3$ and $t\\in [T_1,T]$,\n\\begin{equation}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\tilde\ny^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T |\\tilde z^{m,n}_s|^2\\ {\\rm d}s\\right]\n\\leq \\Dis 2\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\tilde y^{m,n}_r|^2 \\right]\\right){\\rm d}s+{1\\over 8}\\E\\left[\\int_t^T |\\tilde z^{m,n-1}_s|^2\\ {\\rm d}s\\right].\n\\end{equation}\nNote by Remark 1 that $\\kappa(x)\\leq A(x+1)$ for each $x\\geq 0$. Gronwall's inequality together with (113) and (98) yields that for each $m\\geq 0$, $n\\geq 3$ and $t\\in [T_1,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\tilde y^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T | \\tilde z^{m,n}_s|^2\\ {\\rm\nd}s\\right]\n&\\leq &\\Dis \\left(2AT+{1\\over 8}\\E\\left[\\int_t^T |\\tilde z^{m,n-1}_s|^2\\ {\\rm d}s\\right]\\right)\\cdot e^{2A(T-t)}\\vspace{0.1cm}\\\\\n&\\leq &\\Dis 4AT+{1\\over 4}\\E\\left[\\int_t^T\n|\\tilde z^{m,n-1}_s|^2\\ {\\rm d}s\\right]\\vspace{0.1cm}\\\\\n&\\leq &\\Dis {16AT\\over 3}+{1\\over 4^{n-2}}\\E\\left[\\int_t^T |\\tilde z^{m,2}_s|^2\\ {\\rm d}s\\right].\n\\end{array}\n\\end{equation}\nFurthermore, combining (109) and (111) with $n=2$, by virtue of Doob's inequality we obtain the existence of a constant $K_{A,T}>0$ depending only on $A$ and $T$ such that for each $m\\geq 0$ and $t\\in [T_1,T]$,\n\\begin{equation}\n\\E\\left[\\int_t^T|\\tilde z^{m,2}_s|^2\\ {\\rm d}s\\right]\\leq K_{A,T}\\left(1+\\E\\left[\\left(\\int_t^T \\tilde \\triangle_s^{m,2}\\ {\\rm d}s\\right)^2\\right]\\right).\n\\end{equation}\nAnd, in view of (106) with $n=2$ and the fact of  $2\\alpha\\in (0,1)$, it follows from H\\\"{o}lder's inequality and Jensen's inequality that for each $m\\geq 0$ and $t\\in [T_1,T]$,\n$$\n\\begin{array}{lll}\n\\Dis \\E\\left[\\left(\\int_t^T \\tilde \\triangle_s^{m,2}\\ {\\rm d}s\\right)^2\\right]\n&\\leq & \\Dis 16\\gamma^2T^{2-2\\alpha}\\left(\\E\\left[\\int_t^T g_s{\\rm d}s\\right]\\right)^{2\\alpha}+16\\gamma^2T^2\\E\\left[\\sup_{s\\in [t,T]}|y^{m,1}_s|^{2\\alpha}\\right]\\\\\n&&\\Dis +16\\gamma^2T^{2-\\alpha}\\E\\left[\\left(\\int_t^T |z^{m,1}_s|^2\\ {\\rm d}s\\right)^{\\alpha}\\right],\n\\end{array}\n$$\nfrom which as well as (67) with $n=1$, we can deduce that\n\\begin{equation}\n\\RE\\ t\\in [T_1,T],\\ \\ \\sup_{m\\geq 0}\\E\\left[\\left(\\int_t^T \\tilde \\triangle_s^{m,2}\\ {\\rm d}s\\right)^2\\right]<+\\infty.\n\\end{equation}\nThus, the inequality (102) follows from (114)-(116).\\vspace{0.1cm}\n\nFinally, combining (101) and (102) we can deduce that for each $t\\in [T_1,T]$,\n\\begin{equation}\n\\sup_{n\\geq 2}\\sup_{m\\geq 0}\\left(\\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\bar y^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T | \\bar z^{m,n}_s|^2\\ {\\rm d}s\\right]\\right)<+\\infty.\n\\end{equation}\nThus, by first taking supremum with respect to $m$ and then taking $\\limsup$ with respect to $n$ in (99) as well as using Fatou's lemma, the monotonicity and continuity of the function $\\kappa(\\cdot)$ and Bihari's inequality, we can get (88). The proof of Proposition 7 is then completed.\\vspace{0.2cm} \\hfill $\\Box$\n\nNow, we come back to the proof of Theorem 4, and only consider the case when the $\\alpha$ in (H3) values in $(0,1\/2)$. Another case can be proved in the same way.\\vspace{0.2cm}\n\nFirstly, in view of (64) and (65) we have, for each $t\\in [T_1,T]$,\n\\begin{equation}\n\\sup_{s\\in [t,T]}\\E\\left[|y_s^m -y_s^0|\\right] \\leq 2\\sup_{m\\geq 0}\\E\\left[\\sup_{s\\in [t,T]}|y_s^{m,n}-y_s^m|\\right]+\\sup_{s\\in [t,T]}\\E\\left[|y_s^{m,n} -y_s^{0,n}|\\right]\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\E\\left[\\sup_{s\\in [t,T]}|y_s^m -y_s^0|^\\beta\\right] \\leq 2\\sup_{m\\geq 0}\\E\\left[\\sup_{s\\in [t,T]}|y_s^{m,n}-y_s^m|^{\\beta}\\right]+\\E\\left[\\sup_{s\\in [t,T]}|y_s^{m,n}-y_s^{0,n}|^{\\beta}\\right]\n\\end{equation}\nas well as\n\\begin{equation}\n\\Dis \\E\\left[\\left(\\int_t^T|z_s^m -z_s^0|^2{\\rm d}s\\right)^\\beta\\right]\\leq \\Dis 2\\sup_{m\\geq 0}\\E\\left[\\left(\\int_t^T|z_s^{m,n}-z_s^m|^2{\\rm d}s\\right)^\\beta\\right]\n+\\E\\left[\\left(\\int_t^T|z_s^{m,n}-z_s^{0,n}|^2{\\rm d}s\\right)^\\beta\\right].\n\\end{equation}\nThen, letting first $m\\To\\infty$ ($n$ being fixed) and then $n\\To\\infty$ in above three inequalities (118)-(120), in view of (66) and (67) in Proposition 6 and (88) in Proposition 7 we obtain that for each $t\\in [T_1,T]$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\sup\\limits_{s\\in [t,T]}\\E\\left[|y_s^m-y_s^0|\\right]=0,\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{s\\in [t,T]} |y_s^m-y_s^0|^\\beta+\\left(\\int_t^T |z_s^m -z_s^0|^2 {\\rm d}s\\right)^{\\beta\/2} \\right]=0.\n\\end{equation}\nIn the sequel, set $T_j:=(T-j\\delta T)\\vee 0$ for each $j\\geq 2$.\nNoticing that the positive real number $\\delta T$ depends only on $\\lam$ and $A$, we can find a minimal integer $N\\geq 1$ such that $T_N=0$. In view of (121) and (122), if $N=1$, then (61) and (62) have been proved. Otherwise, we consider the BSDEs with parameters $(y^m_{T_1},T_1,g^m)$ and $(y_{T_1},T_1,g)$ defined on the time interval $[T_2,T_1]$. Note by (121) that\n$$\\lim\\limits_{m\\To\\infty} \\E\\left[|y^m_{T_1}-y_{T_1}|\\right]=0.$$\nRepeating the above arguments will yield that for each $t\\in [T_2,T_1]$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\sup\\limits_{s\\in [t,T_1]}\\E\\left[|y_s^m-y_s^0|\\right]=0,\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{s\\in [t,T_1]} |y_s^m-y_s^0|^\\beta+\\left(\\int_t^{T_1} |z_s^m -z_s^0|^2 {\\rm d}s\\right)^{\\beta\/2} \\right]=0.\n\\end{equation}\nThus, in view of (121)-(124), if $N=2$, then (61) and (62) have also been proved. Otherwise, we can consider successively the BSDEs on $[T_3,T_2]$, $\\cdots$, $[0,T_{N-1}]$, and finally complete the proof of Theorem 4 by repeating the above procedure.\\hfill $\\Box$\\vspace{0.2cm}\n\n{\\bf Remark 6}\\ \\ The whole idea of the proof of  Theorem 4 is involved in, by virtue of (63), introducing $(y_\\cdot^{m,n},z_\\cdot^{m,n})$ as a bridge between $(y_\\cdot^m,z_\\cdot^m)$ and $(y_\\cdot^0,z_\\cdot^0)$, and then, by virtue of (118)-(120), proving Propositions 6 and 7 respectively. This whole idea should be  new. In addition, Propositions 6 and 7 are not easy to prove, especially a delicate argument has been done to obtain the inequality (117).\\vspace{0.3cm}\n\nIn the sequel, we will investigate the stability theorem of the solutions in the space $\\s^1\\times {\\rm M}^1$ for multidimensional BSDEs.\\vspace{0.2cm}\n\nNow, for each $m\\in \\N$, let $\\xi^m\\in \\LT$ and let  $(y_t^m, z_t^m)_{t\\in [0,T]}$ be\na solution in $\\s^1\\times {\\rm M}^1$ for\nthe following BSDEs depending on parameter $m$:\n\\begin{equation}\ny_t^m=\\xi^m+\\int_t^T g^m(s,y_s^m,z_s^m)\\ {\\rm d}s -\\int_t^T z_s^m{\\rm d}B_s,\\quad t\\in [0,T].\n\\end{equation}\nFurthermore,  we introduce  the following assumptions:\\vspace{0.2cm}\n\n{\\bf (B3)}\\ All $g^m$ and $\\xi^m$ satisfy assumptions (H1)-(H4) with the same parameters $\\rho(\\cdot)$, $\\lambda$, $\\gamma$, $g_t$ and $\\alpha$. Furthermore, in the case of $\\alpha\\in [1\/2,1)$ we assume that $g$ satisfies (H1) with a function $\\rho(\\cdot)$ satisfying (14) for some $\\bar p>1$. \\vspace{0.2cm}\n\n{\\bf (B4)}\\ There exists a nonnegative real number sequence\n$\\{a_m\\}_{m=1}^{+\\infty}$ satisfying $\\lim\\limits_{m\\To \\infty}a_m=0$ such that $\\as$,  for each $m\\geq 1$,\\vspace{-0.1cm}\n\\begin{equation}\n\\RE\\ (y,z)\\in \\R^k\\times \\R^{k\\times d},\\ \\ |g^m(\\omega,t,y,z)-g^0(\\omega,t,y,z)|\\leq a_m.\n\\end{equation}\nAnd\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup_{t\\in\\T}\\E\\left[\\left.\n|\\xi^{m}-\\xi^{0}|\\right|\\F_t\\right]\\right]=0.\\vspace{0.2cm}\n\\end{equation}\n\nThe following Theorem 5 is the stability theorem of the solutions in $\\s^1\\times {\\rm M}^1$.\\vspace{0.1cm}\n\n{\\bf Theorem 5}\\ Under assumptions (B3) and (B4), we have\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^m-y_t^0|+\\left(\\int_0^T |z_s^m -z_s^0|^2\\  {\\rm d}s\\right)^{1\/2} \\right]=0.\\vspace{0.1cm}\n\\end{equation}\n\n{\\bf Proof}\\ \\ First of all, it follows from (62) of Theorem 4 that the sequence of variables $\\{\\sup_{t\\in [0,T]} |y_t^m-y_t^0|\\}_{m=1}^{+\\infty}$ converges in probability to $0$ as $m\\To\\infty$. In the sequel, we will first prove that it is also uniformly integrable, and then\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^m-y_t^0|\\right]=0.\\vspace{0.2cm}\n\\end{equation}\nIn fact, set $\\hat y^m_\\cdot:=y^m_\\cdot-y^0_\\cdot$ and $\\hat z^m_\\cdot:=z^m_\\cdot-z^0_\\cdot$. Then, for each $m\\geq 1$, $(\\hat y^m_t,\\hat z^m_t)_{t\\in \\T}$ is a solution in the space $\\s^1\\times {\\rm M}^1$ for the following BSDE:\n\\begin{equation}\n\\hat{y}_t^m=\\xi^{m}-\\xi^0+\\int^{T}_{t}\\hat g^m(s,\\hat y_s^m,\\hat z_s^m)\\ \\mathrm{d} s-\\int^{T}_{t}\\hat{z}_s^m\\ \\mathrm{d}B_s, \\quad t\\in\\T,\n\\end{equation}\nwhere for each $y\\in\\R^k$,\n$\\hat g^m(t,y,z):=g^m(t,y+y^0_t,z+z^0_t)\n-g^0(t,y^0_t,z^0_t).$\nIt follows from assumptions (H1) and (H3) of $g^m$ together with (126) and Remark 1 that $\\as$, for each $y\\in\\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\hat{g}^m(t,y,z)\\rangle &\\leq &\\langle y,g^m(t,y+y_t^0,z+z^0_t)-g^m(t,y_t^0,z+z^0_t)\n\\rangle+|y||g^m(t,y_t^0,z+z^0_t)-g^m(t,y_t^0,z^0_t)|\\\\\n&& \\Dis +|y||g^m(t,y_t^0,z^0_t)-g^0(t,y_t^0,z^0_t)|\\\\\n&\\leq & \\Dis \\kappa(|y|^2)+\\lam |y||z|+|y|a_m\n\\end{array}\n\\end{equation}\nand\n\\begin{equation}\n\\left\\langle{\\hat{y}^m_t\\over |\\hat{y}^m_t|} \\mathbbm{1}_{|\\hat{y}^m_t|\\neq 0},g^m(t,y^m_t,z^m_t)\n-g^0(s,y_t^0,z_t^0)\\right\\rangle\\leq \\rho(|\\hat{y}^m_t|)+\\hat\\triangle_t^m+a_m,\\vspace{0.2cm}\n\\end{equation}\nwhere $\\kappa(\\cdot)$ is defined in (H1a)$_2$, and\n\\begin{equation}\n\\hat\\triangle^m_t:=|g^m(t,y_t^0,z^m_t)-g^m(t,y_t^0,z^0_t)| \\leq 2\\gamma \\left(g_t+\n|y_t^0|+|z^m_t|+|z^0_t|\\right)^\\alpha.\n\\end{equation}\nNext, in view of (132) and Remark 1, using Corollary 2.3 in \\cite{Bri03} and Gronwall's inequality we get that for each $m\\geq 1$ and $t\\in\\T$,\n\\begin{equation}\n|\\hat y_t^m|\\leq \\left(\\E\\left[\\left.|\\xi^m-\\xi^0|\\right|\\F_t\n\\right]+AT+\\E\\left[\\left.\\int_0^T \\hat\\triangle^m_s\\ {\\rm d}s\\right|\\F_t\\right]+a_mT\\right)\\cdot e^{AT}.\n\\end{equation}\nOn one hand, it follows from (127) that $\\{\\sup_{t\\in\\T}\\E\\left[\\left.\n|\\xi^{m}-\\xi^{0}|\\right|\\F_t\\right]\\}_{m=1}^{+\\infty}$ is uniformly integrable. On the other hand, in view of (133), for each $q>1$ such that $\\alpha q<1$, H\\\"{o}lder's inequality yields the existence of a positive constant $C$ independent of $m$ such that\n\\begin{equation}\n\\E\\left[\\left(\\int_0^T \\hat\\triangle^m_s\\ {\\rm d}s\\right)^q\\right]<C\\left(1+\\E\\left[\\left(\\int_0^T |z^m_s|^2\\ {\\rm d}s\\right)^{\\alpha q\\over 2} \\right]\\right).\n\\end{equation}\nAnd, Doob's inequality together with (135) and (62) leads to\n\\begin{equation}\n\\sup_{m\\geq 1}\\E\\left[\\sup_{t\\in\\T}\\left( \\E\\left[\\left.\\int_0^T \\hat\\triangle^m_s\\ {\\rm d}s\\right|\\F_t\\right]\\right)^q\\right]\\leq\n\\sup_{m\\geq 1}\\E\\left[\\left(\\int_0^T \\hat\\triangle^m_s\\ {\\rm d}s\\right)^q\\right]<+\\infty,\n\\end{equation}\nwhich means that the sequence of variables $$\\left\\{\\sup_{t\\in\\T}\\E\\left[\\left.\\int_0^T \\hat\\triangle^m_s\\ {\\rm d}s\\right|\\F_t\\right]\\right\\}_{m=1}^{+\\infty}$$\nis uniformly integrable. Thus, in view of (134) and (B4), we can deduce that the sequence of variables $\\{\\sup_{t\\in [0,T]} |\\hat y_t^m|\\}_{m=1}^{+\\infty}$ is also uniformly integrable, and get (129).\\vspace{0.1cm}\n\nFinally, in view of (131), Remark 1, Lemma 1 and (129), using Proposition 1 with $p=1$ and a similar argument to that from (30) to (33) we can obtain that \\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\left(\\int_0^T |z_s^m -z_s^0|^2\\  {\\rm d}s\\right)^{1\/2} \\right]=0.\n\\end{equation}\nThus, Theorem 5 is proved by (129) and (137).\\vspace{0.3cm} \\hfill $\\Box$\n\n{\\bf Remark 7}\\ We mention that Theorem 5 can also be proved in the same way as Theorem 4. In fact, it can be proved that (67) holds for $\\beta=1$ under the assumptions (B1) and (B2) of Theorem 5.\\vspace{0.2cm}\n\nBy Theorems 4 and 5, the following corollary follows immediately.\\vspace{0.1cm}\n\n{\\bf Corollary 2}\\ \\ Assume that the generator $g$ satisfies assumptions (H1)-(H3). In the case when the $\\alpha$ defined in (H3) values in $[1\/2,1)$, we also assume that there exists a constant $\\bar p>1$ such that the function $\\rho(\\cdot)$ in (H1) satisfies (14). Suppose that for each $m\\geq 1$, $\\xi^m,\\xi\\in \\LT$ and $(y^m_t,z^m_t)_{t\\in\\T}$ and $(y_t,z_t)_{t\\in\\T}$ are respectively the unique $L^1$ solution of BSDE$(\\xi^m,T,g)$ and BSDE$(\\xi,T,g)$. If $\\lim\\limits_{m\\To\\infty}\\E\\left[|\\xi^m-\\xi|\\right]=0,$ then\n$$\n\\lim\\limits_{m\\To \\infty}\\sup\\limits_{t\\in [0,T]}\\E\\left[|y_t^m-y_t|\\right]=0,\n$$\nand for each $\\beta\\in (0,1)$,\n$$\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^m-y_t|^\\beta+\\left(\\int_0^T |z_s^m -z_s|^2 {\\rm d}s\\right)^{\\beta\/2} \\right]=0.\n$$\nMoreover, if for each $m\\geq 1$, $\\xi^m$ and $g(t,0,0)$ satisfy assumptions (H4), and\n$$\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup_{t\\in\\T}\\E\\left[\\left.\n|\\xi^{m}-\\xi|\\right|\\F_t\\right]\\right]=0,\n$$\nthen\n$$\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^m-y_t|+\\left(\\int_0^T |z_s^m -z_s|^2\\  {\\rm d}s\\right)^{1\/2} \\right]=0.\\vspace{0.3cm}\n$$\n\n{\\bf Remark 8}\\ By Remarks 5 and 2, we know that Theorems 4-5 and Corollary 2 give the stability of the $L^1$ solutions of multidimensional BSDEs investigated in \\citet{Bri03}.\\vspace{0.1cm}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Gravito-electromagnetic analogy based on tidal tensors}\n\nThe topic of the gravito-electromagnetic analogies has a long story,\nwith different analogies being unveiled throughout the years. Some\nare purely formal analogies, like the splitting of the Weyl tensor\nin electric and magnetic parts, e.g. \\cite{Maartens}; but others\n(e.g \\cite{DSX,CostaHerdeiro2008,Jantzen,Natario,Ruggiero:2002})\nstem from certain physical similarities between the gravitational\nand electromagnetic interactions. The linearized Einstein equations\n(see e.g. \\linebreak  \\cite{DSX,Ruggiero:2002,Gravitation and Inertia}), in the\nharmonic gauge $\\bar{h}_{\\alpha\\beta}^{\\,\\,\\,\\,\\,\\,,\\beta}=0$, take\nthe form $\\square\\bar{h}^{\\alpha\\beta}=-16\\pi T^{\\alpha\\beta}\/c^{4}$,\nsimilar to Maxwell equations in the Lorentz gauge: $\\square A^{\\beta}=-4\\pi j^{\\beta}\/c$.\nThat suggests an analogy between the trace reversed time components\nof the metric tensor $\\bar{h}_{0\\alpha}$ and the electromagnetic\n4-potential $A_{\\alpha}$. Defining the 3-vectors usually dubbed gravito-electromagnetic\nfields, the time components of these equations may be cast in a Maxwell-like\nform, e.g. eqs (16)-(22) of \\cite{Ruggiero:2002}. Furthermore (on\ncertain special conditions, see section \\ref{sec:Linearized-Gravity}) \\linebreak\ngeodesics, precession and forces on gyroscopes are described in terms\nof these fields in a \\linebreak form similar to their electromagnetic counterparts,\ne.g. \\cite{Ruggiero:2002}, \\linebreak \\cite{Gravitation and Inertia}. Such analogy may\nactually be cast in an exact form using the 3+1 splitting of spacetime\n(see \\cite{Jantzen,Natario}).\n\nThese are analogies comparing physical quantities (electromagnetic\nforces) from one theory with inertial gravitational forces (i.e. fictitious\nforces, that can be gauged away by moving to a freely falling frame,\ndue to the equivalence principle); it is clear that (non-spinning)\ntest particles in a gravitational field move with zero acceleration\n$DU^{\\alpha}\/d\\tau=0$; and that the spin 4-vector of a gyroscope\nundergoes Fermi-Walker transport $DS^{\\alpha}\/d\\tau=S_{\\sigma}U^{\\alpha}DU^{\\sigma}\/d\\tau$,\nwith no real torques applied on it. In this sense the gravito-electromagnetic\nfields are pure coordinate artifacts, attached to the observer's frame.\n\nHowever, these approaches describe also (not through the {}``gravito-electromagnetic''\nfields themselves, but through their derivatives; and, again, under\nvery special conditions) tidal effects, like the force applied on\na gyroscope. And these are covariant effects, implying physical gravitational\nforces. \n\nHerein we will discuss under which precise conditions a similarity\nbetween gravity and electromagnetism occurs (that is, under which\nconditions the physical analogy $\\bar{h}_{0\\mu}\\leftrightarrow A_{\\mu}$\nholds, and Eqs. like (16)-(22) of \\cite{Ruggiero:2002} have a physical\ncontent). For that we will make use of the tidal tensor formalism\nintroduced in \\cite{CostaHerdeiro2008}. The advantage of this formalism\nis that, by contrast with the approaches mentioned above, it is based\non quantities which can be covariantly defined in both theories ---\ntidal forces (the only physical forces present in gravity) --- which\nallows for a more transparent comparison between the electromagnetic\n(EM) and gravitational (GR) interactions.\n\n\\begin{table}[h]\n\\caption{\\label{analogy}The gravito-electromagnetic analogy based on tidal\ntensors. }\n\n\n\\begin{centering}\n\\setlength{\\arrayrulewidth}{0.8pt}\\begin{tabular}{>{\\centering}p{39.6ex}c>{\\centering}p{41.9ex}c}\n\\hline \n\\multicolumn{2}{c}{\\raisebox{3ex}{}\\raisebox{0.5ex}{Electromagnetism}} & \\multicolumn{2}{c}{\\raisebox{3ex}{}\\raisebox{0.5ex}{Gravity}}\\tabularnewline\n\\hline \n\\raisebox{3ex}{}Worldline deviation:  &  & \\raisebox{3ex}{}Geodesic deviation:  & \\tabularnewline\n\\raisebox{6ex}{}\\raisebox{2ex}{${\\displaystyle \\frac{D^{2}\\delta x^{\\alpha}}{d\\tau^{2}}=\\frac{q}{m}E_{\\,\\,\\,\\beta}^{\\alpha}\\delta x^{\\beta}},\\,\\,\\, E_{\\,\\,\\,\\beta}^{\\alpha}\\equiv F_{\\ \\mu;\\beta}^{\\alpha}U^{\\mu}$}  & \\raisebox{2ex}{(1a) \\,\\,}  & \\raisebox{5.5ex}{}\\raisebox{2ex}{${\\displaystyle \\frac{D^{2}\\delta x^{\\alpha}}{d\\tau^{2}}=-\\mathbb{E}_{\\,\\,\\,\\beta}^{\\alpha}\\delta x^{\\beta}},\\,\\,\\,\\mathbb{E}_{\\,\\,\\,\\beta}^{\\alpha}\\equiv R_{\\,\\,\\,\\mu\\beta\\nu}^{\\alpha}U^{\\mu}U^{\\nu}$}  & \\raisebox{2ex}{(1b)}\\tabularnewline\n\\hline \n\\raisebox{3ex}{}Force on magnetic dipole:  &  & \\raisebox{3ex}{}Force on gyroscope:  & \\tabularnewline\n\\raisebox{6ex}{}\\raisebox{2ex}{${\\displaystyle F_{EM}^{\\beta}=\\frac{q}{2m}B_{\\alpha}^{\\,\\,\\,\\beta}S^{\\alpha}},\\,\\,\\, B_{\\,\\,\\,\\beta}^{\\alpha}\\equiv\\star F_{\\ \\mu;\\beta}^{\\alpha}U^{\\mu}$}  & \\raisebox{2ex}{(2a)}  & \\raisebox{6ex}{}\\raisebox{2ex}{~~${\\displaystyle F_{G}^{\\beta}=-\\mathbb{H}_{\\alpha}^{\\,\\,\\,\\beta}S^{\\alpha}},\\,\\,\\,\\mathbb{H}_{\\,\\,\\,\\beta}^{\\alpha}\\equiv\\star R_{\\,\\,\\,\\mu\\beta\\nu}^{\\alpha}U^{\\mu}U^{\\nu}$}  & \\raisebox{2ex}{(2b)}\\tabularnewline\n\\hline \n\\raisebox{3ex}{}Maxwell Equations:  &  & \\raisebox{3ex}{}Eqs. Grav. Tidal Tensors:  & \\tabularnewline\n\\raisebox{3ex}{}$E_{\\,\\,\\,\\alpha}^{\\alpha}=4\\pi\\rho_{c}$  & (3a)  & \\raisebox{3ex}{}$\\mathbb{E}_{\\,\\,\\,\\alpha}^{\\alpha}=4\\pi\\left(2\\rho_{m}+T_{\\,\\,\\alpha}^{\\alpha}\\right)$  & (3b)\\tabularnewline\n\\raisebox{3.5ex}{}$E_{[\\alpha\\beta]}=\\frac{1}{2}F_{\\alpha\\beta;\\gamma}U^{\\gamma}$  & (4a)  & \\raisebox{3.5ex}{}$\\mathbb{E}_{[\\alpha\\beta]}=0$  & (4b)\\tabularnewline\n\\raisebox{3.5ex}{}$B_{\\,\\,\\,\\alpha}^{\\alpha}=0$  & (5a)  & \\raisebox{3.5ex}{}$\\mathbb{H}_{\\,\\,\\,\\alpha}^{\\alpha}=0$  & (5b)\\tabularnewline\n\\raisebox{5.5ex}{}\\raisebox{2ex}{$B_{[\\alpha\\beta]}=\\frac{1}{2}\\star F_{\\alpha\\beta;\\gamma}U^{\\gamma}-2\\pi\\epsilon_{\\alpha\\beta\\sigma\\gamma}j^{\\sigma}U^{\\gamma}$}  & \\raisebox{2ex}{(6a)}  & ~~\\raisebox{5.5ex}{}\\raisebox{2ex}{$\\mathbb{H}_{[\\alpha\\beta]}=-4\\pi\\epsilon_{\\alpha\\beta\\sigma\\gamma}J^{\\sigma}U^{\\gamma}$}~~  & \\raisebox{2ex}{(6b)}\\tabularnewline\n\\hline\n\\end{tabular}\\\\\n \n\\par\\end{centering}\n\n\\begin{raggedright}\n{\\scriptsize $\\rho_{c}=-j^{\\alpha}U_{\\alpha}$ and $j^{\\alpha}$ are,\nrespectively, the charge density and current 4-vector; $\\rho_{m}=T_{\\alpha\\beta}U^{\\alpha}U^{\\beta}$\nand $J^{\\alpha}=-T_{\\,\\beta}^{\\alpha}U^{\\beta}$ are the mass\/energy\ndensity and current (quantities measured by the observer of 4-velocity\n$U^{\\alpha}$); $T_{\\alpha\\beta}\\equiv$ energy-momentum tensor; $S^{\\alpha}\\equiv$\nspin 4-vector; $\\star\\equiv$ Hodge dual. We use $\\tilde{e}_{0123}=-1$.}\n\\par\\end{raggedright}\n\\end{table}\n\n\nThe tidal tensor formalism unveils a new gravito-electromagnetic analogy,\nsummarized in Table \\ref{analogy}, based on exact and covariant equations.\nThese equations make clear key differences, and under which conditions\na similarity between the two interactions may occur.\n\nEqs. (1) are the worldline deviation equations yielding the relative\nacceleration of two neighboring particles (connected by the infinitesimal\nvector $\\delta x^{\\alpha}$) with the \\textit{same} 4-velocity $U^{\\alpha}$\n(and the same $q\/m$ ratio, in the electromagnetic case). These equations\nmanifest the physical analogy between electric tidal tensors: $\\mathbb{E}_{\\alpha\\beta}\\leftrightarrow E_{\\alpha\\beta}$.\n\nEq. (2a) yields the electromagnetic force exerted on a magnetic dipole\nmoving with 4-velocity $U^{\\alpha}$, and is the covariant generalization\nof the usual 3-D expression $\\mathbf{F_{EM}}=\\nabla(\\mathbf{S}.\\mathbf{B})q\/2m$\n(valid only in the dipole's proper frame); Eq. (2b) is exactly the\nPapapetrou-Pirani equation for the gravitational force exerted on\na spinning test particle. In both (2a) and (2b), Pirani's supplementary\ncondition $S_{\\mu\\nu}U^{\\nu}=0$ is assumed (c.f. \\cite{CostaHerdeiro2009}).\nThese equations manifest the physical analogy between magnetic tidal\ntensors: $B_{\\alpha\\beta}\\leftrightarrow\\mathbb{H}_{\\alpha\\beta}$.\n\nTaking the traces and antisymmetric parts of the EM tidal tensors,\none obtains Eqs. (3a)-(6a), which are explicitly covariant forms for\neach of Maxwell equations. Eqs. (3a) and (6a) are, respectively, the\ntime and space projections of Maxwell equations $F_{\\ \\ \\ ;\\beta}^{\\alpha\\beta}=4\\pi j^{\\alpha}$;\ni.e., they are, respectively, covariant forms of $\\nabla\\cdot\\mathbf{E}=4\\pi\\rho_{c}$\nand $\\nabla\\times\\mathbf{B}=\\partial\\mathbf{E}\/\\partial t+4\\pi\\mathbf{j}$;\nEqs. (4a) and (5a) are the space and time projections of the electromagnetic\nBianchi identity $\\star F_{\\ \\ \\ ;\\beta}^{\\alpha\\beta}=0$; i.e.,\nthey are covariant forms for $\\nabla\\times\\mathbf{E}=-\\partial\\mathbf{B}\/\\partial t$\nand $\\nabla\\cdot\\mathbf{B}=0$. These equations involve only tidal\ntensors and sources, which can be seen substituting the following\ndecomposition (or its Hodge dual) in (4a) and (6a): \\begin{equation}\nF_{\\alpha\\beta;\\gamma}=2U_{[\\alpha}E_{\\beta]\\gamma}+\\epsilon_{\\alpha\\beta\\mu\\sigma}B_{\\,\\,\\,\\gamma}^{\\mu}U^{\\sigma}\\\n.\\label{Fdecomp}\\end{equation}\nIt is then straightforward to obtain the \\emph{physical} gravitational\nanalogues of Maxwell equations: one just has to apply the same procedure\nto the gravitational tidal tensors, i.e., write the equations for\ntheir traces and antisymmetric parts (that is more easily done decomposing\nthe Riemann tensor in terms of the Weyl tensor and source terms, see\n\\cite{CostaHerdeiro2007} sec. 2), which leads to Eqs. (3b) - (6b).\nUnderlining the analogy with the situation in electromagnetism, Eqs.\n(3b) and (6b) turn out to be the time-time and and time-space projections\nof Einstein equations $R_{\\mu\\nu}=8\\pi(T_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu}T_{\\,\\,\\,\\alpha}^{\\alpha})$,\nand Eqs. (4b) and (5b) the time-space and time-time projections of\nthe algebraic Bianchi identities $\\star R_{\\ \\ \\ \\gamma\\beta}^{\\gamma\\alpha}=0$.\n\n\n\\subsection{Gravity vs Electromagnetism}\n\n\\emph{Charges} --- the gravitational analogue of $\\rho_{c}$ is $2\\rho_{m}+T_{\\,\\,\\alpha}^{\\alpha}$\n($\\rho_{m}+3p$ for a perfect fluid) $\\Rightarrow$ in gravity, pressure\nand all material stresses contribute as sources.\n\n\\emph{Ampere law} --- in stationary (in the observer's rest frame)\nsetups, $\\star F_{\\alpha\\beta;\\gamma}U^{\\gamma}$ vanishes and equations\n(6a) and (6b) match up to a factor of 2 $\\Rightarrow$ currents of\nmass\/energy source gravitomagnetism like currents of charge source\nmagnetism.\n\n\n\\emph{Symmetries of Tidal Tensors} --- The GR and EM tidal tensors\ndo not generically exhibit the same symmetries, signaling fundamental\ndifferences between the two interactions. In the general case of fields\nthat are time dependent in the observer's rest frame (that is the\ncase of an intrinsically non-stationary field, or an observer moving\nin a stationary field), the electric tidal tensor $E_{\\alpha\\beta}$\npossesses an antisymmetric part, which is the covariant derivative\nof the Maxwell tensor along the observer's worldline; there is also\nan antisymmetric contribution $\\star F_{\\alpha\\beta;\\gamma}U^{\\gamma}$\nto $B_{\\alpha\\beta}$. These terms consist of time projections of\nEM tidal tensors (cf. decomposition \\ref{Fdecomp}), and contain the\nlaws of electromagnetic induction. The gravitational tidal tensors,\nby contrast, are symmetric (in vacuum, in the magnetic case) and spatial,\nmanifesting the absence of analogous effects in gravity.\n\n\\emph{Gyroscope vs. magnetic dipole} --- According to Eqs. (2), both\nin the case of the magnetic dipole and in the case of the gyroscope,\nit is the magnetic tidal tensor, \\emph{as seen by the test particle}\n($U^{\\alpha}$ in Eqs. (2) is the gyroscope\/dipole 4-velocity), that\ndetermines the force exerted upon it. Hence, from Eqs. (6), we see\nthat the forces can be similar only if the fields are stationary (besides\nweak) in the gyroscope\/dipole frame, i.e., when it is at {}``rest''\nin a stationary field. Eqs. (2) also tell us that in gravity the angular\nmomentum $S$ plays the role of the magnetic moment $\\mu=S(q\/2m)$;\nthe relative minus sign manifests that masses\/charges of the same\nsign attract\/repel one another in gravity\/electromagnetism, as do\ncharge\/mass currents with parallel velocity. \n\n\n\\section{Linearized Gravity\\label{sec:Linearized-Gravity}}\n\nIf the fields are stationary in the observer's rest frame, the GR\nand EM tidal tensors have the same symmetries, which by itself does\nnot mean a close similarity between the two interactions (note that\ndespite the analogy in Table 1, EM tidal tensors are linear, whereas\nthe GR ones are not). But in two special cases a matching between\ntidal tensors occurs: ultrastationary spacetimes (where the gravito-magnetic\ntidal tensor is linear, see \\cite{CostaHerdeiro2008} Sec. IV) and\nlinearized gravitational perturbations, which is the case of interest\nfor astronomical applications. \n\nConsider an arbitrary electromagnetic field $A^{\\alpha}=(\\phi,\\mathbf{A})$\nand arbitrary perturbations around Minkowski spacetime in the form%\n\\footnote{In the previous sections we were putting $c=1$. In this section we\nre-introduce the speed of light in order to facilitate comparison\nwith relevant literature.%\n}\n\n\\begin{equation}\nds^{2}=-c^{2}\\left(1-2\\frac{\\Phi}{c^{2}}\\right)dt^{2}-\\frac{4}{c}\\mathcal{A}_{j}dtdx^{j}+\\left[\\delta_{ij}+2\\frac{\\Theta_{ij}}{c^{2}}\\right]dx^{i}dx^{j}\\ .\\label{Linear pert}\\end{equation}\n\n\n\\textbf{Tidal effects.} --- The GR and EM tidal tensors from these\nsetups will be in general very different, as is clear from equations\n(3-6), and as one may check from the explicit expressions in \\cite{CostaHerdeiro2008}.\n\nBut if one considers time independent fields, and a static observer\nof 4-velocity $U^{\\mu}=c\\delta_{0}^{\\mu}$, then the \\emph{linearized}\ngravitational tidal tensors match their electromagnetic counterparts\nidentifying $(\\phi,A^{i})\\leftrightarrow(\\Phi,\\mathcal{A}^{i})$ (in\nexpressions below colon represents partial derivatives; $\\epsilon_{ijk}\\equiv$\nLevi Civita symbol): \\begin{equation}\n\\mathbb{E}_{ij}\\simeq-\\Phi_{,ij}\\stackrel{\\Phi\\leftrightarrow\\phi}{=}E_{ij},\\quad\\mathbb{H}_{ij}\\simeq\\epsilon_{i}^{\\,\\,\\, lk}\\mathcal{A}_{k,lj}\\stackrel{\\mathcal{A}\\leftrightarrow A}{=}B_{ij}\\ .\\label{MatchingLinear}\\end{equation}\n This suggests the physical analogy $(\\phi,A^{i})\\leftrightarrow(\\Phi,\\mathcal{A}^{i})$,\nand defining the {}``gravito-electro-magnetic fields'' $\\mathbf{E_{G}}=-\\nabla\\Phi$\nand $\\mathbf{B_{G}}=\\nabla\\times\\bm{\\mathcal{A}}$, in analogy with\nthe electromagnetic fields $\\mathbf{E}=-\\nabla\\phi,$ $\\mathbf{B}=\\nabla\\times\\mathbf{A}$.\nIn terms of these fields we have $\\mathbb{E}_{ij}\\simeq(E_{G})_{i,j}$\nand $\\mathbb{H}_{ij}\\simeq(B_{G})_{i,j}$, in analogy with the electromagnetic\ntidal tensors $E_{ij}=E_{i,j}$ and $B_{ij}=B_{i,j}$. \n\nThe matching (\\ref{MatchingLinear}) means that a gyroscope at rest\n(relative to the static observer) will feel a force $F_{G}^{\\alpha}$\nsimilar to the electromagnetic force $F_{EM}^{\\alpha}$ on a magnetic\ndipole, which in this case take the very simple forms (time components\nare zero): \\begin{equation}\n\\mathbf{F_{EM}}=\\frac{q}{2mc}\\nabla(\\mathbf{B}.\\mathbf{S});\\ \\ \\ \\ \\ F_{G}^{j}=-\\frac{1}{c}\\mathbb{H}^{ij}S_{i}\\approx-\\frac{1}{c}(B_{G})^{i,j}S_{i}\\ \\Leftrightarrow\\ \\mathbf{F_{G}}=-\\frac{1}{c}\\nabla(\\mathbf{B_{G}}.\\mathbf{S})\\ .\\label{FG_Stationary}\\end{equation}\nHad we considered gyroscopes\/dipoles with different 4-velocities,\nnot only the expressions for the forces would be more complicated,\nbut also the gravitational force would significantly differ from the\nelectromagnetic one, as one may check comparing Eqs. (12) with (17)-(20)\nof \\cite{CostaHerdeiro2008}. This will be exemplified in section\n\\ref{Translational-vs.-Rotational}.\n\nThe matching (\\ref{MatchingLinear}) also means, by similar arguments,\nthat the relative acceleration between two neighboring masses $D^{2}\\delta x^{i}\/d\\tau^{2}=-\\mathbb{E}^{ij}\\delta x_{j}$\nis similar to the relative acceleration between two charges (with\nthe same $q\/m$): $D^{2}\\delta x^{i}\/d\\tau^{2}=E^{ij}\\delta x_{j}(q\/m)$,\n\\emph{at the instant} when the test particles have 4-velocity $U^{\\alpha}=c\\delta_{0}^{\\alpha}$\n(i.e., are \\emph{at rest} relative to the static observer $\\mathcal{O}$).\n\n\\textbf{Gyroscope precession.} --- The evolution of the spin vector\nof the gyroscope is given by the Fermi-Walker transport law, which,\nfor a gyroscope at rest reads $DS^{i}\/d\\tau=0$; hence, we have, in\nthe coordinate basis, Eq. (\\ref{PrecessGen}a). The last term of Eq.\n(\\ref{PrecessGen}a) vanishes if we express $\\mathbf{S}$ in the local\northonormal tetrad $e^{\\hat{\\alpha}}$: $S^{i}=S^{\\hat{i}}e_{\\,\\hat{i}}^{i}$,\nwhere to linear order $e_{\\,\\hat{i}}^{i}=\\delta_{\\ \\hat{i}}^{i}-\\Theta_{\\ \\ \\hat{i}}^{i}\/c^{2}$;\nin this fashion we obtain Eq. (\\ref{PrecessGen}b), which is similar\nto the precession of a magnetic dipole in a magnetic field $d\\mathbf{S}\/dt=q\\mathbf{S}\\times\\mathbf{B}\/2mc$:\n\\begin{equation}\n\\frac{dS^{i}}{dt}=-c\\Gamma_{0j}^{i}S^{j}=-\\frac{1}{c}\\left[(\\mathbf{S}\\times\\mathbf{B_{G}})^{i}+\\frac{1}{c}\\frac{\\partial\\Theta_{\\ }^{ij}}{\\partial t}S_{j}\\right]\\ \\ (a);\\ \\ \\ \\ \\ \\ \\frac{dS^{\\hat{i}}}{dt}=-\\frac{1}{c}(\\mathbf{S}\\times\\mathbf{B_{G}})^{\\hat{i}}\\ \\ (b).\\label{PrecessGen}\\end{equation}\n Thus, in the special case of gyroscope precession, the linear gravito-electromagnetic\nanalogy holds even if the fields vary with time.\n\n\\textbf{Geodesics.} --- The space part of the equation of geodesics\n$U_{\\ ,\\beta}^{\\alpha}U^{\\beta}=-\\Gamma_{\\beta\\gamma}^{\\alpha}U^{\\beta}U^{\\gamma}$\nis given, to first order in the perturbations and in test particle's\nvelocity, by ($a^{i}\\equiv d^{2}x^{i}\/dt^{2})$:\\begin{eqnarray}\n\\mathbf{a} & = & \\nabla\\Phi+\\frac{2}{c}\\frac{\\partial\\bm{\\mathcal{A}}}{\\partial t}-\\frac{2}{c}\\mathbf{v}\\times(\\nabla\\times\\bm{\\mathcal{A}})-\\frac{1}{c^{2}}\\left[\\frac{\\partial\\Phi}{\\partial t}\\mathbf{v}+2\\frac{\\partial\\Theta_{\\ j}^{i}}{\\partial t}v^{j}\\mathbf{e_{i}}\\right]\\ .\\label{geoGeneral}\\end{eqnarray}\nComparing with the electromagnetic Lorentz force:\\begin{equation}\n\\mathbf{a}=\\frac{q}{m}\\left[-\\nabla\\phi-\\frac{1}{c}\\frac{\\partial\\mathbf{A}}{\\partial t}+\\frac{\\mathbf{v}}{c}\\times(\\nabla\\times\\mathbf{A})\\right]=\\frac{q}{m}\\left[\\mathbf{E}+\\frac{\\mathbf{v}}{c}\\times\\mathbf{B}\\right]\\,,\\label{Lorentz}\\end{equation}\nthese equations do not manifest, in general, a close analogy. Note\nthat the last term of (\\ref{geoGeneral}), which has no electromagnetic\nanalogue, is, for the problem at hand (see next section), of the same\norder of magnitude as the second and third terms. But when one considers\nstationary fields, then (\\ref{geoGeneral}) takes the form $\\mathbf{a}=-\\mathbf{E_{G}}-2\\mathbf{v}\\times\\mathbf{B_{G}}\/c$\nanalogous to (\\ref{Lorentz}).\n\nNote the difference between this analogy and the one from the tidal\neffects considered above: in the case of the latter, the similarity\noccurs only when the \\emph{test particle} sees time independent \\emph{fields}\n(fields $\\equiv$ derivatives of potentials\/of metric perturbations);\nfor geodesics, it is when \\emph{the observer} (not the test particle!)\nsees a time independent \\emph{potential }($\\phi$)\\emph{\/metric perturbations}($\\Phi,\\Theta_{ij}$).\n\n\n\\subsection{\\label{Translational-vs.-Rotational}Translational vs. Rotational\nMass Currents}\n\nThe existence of a similarity between gravity and electromagnetism\nthus relies on the time dependence of the mass currents: if the currents\nare (nearly) stationary, for instance from a spinning celestial body,\nthe gravitational field generated is analogous to a magnetic field;\nan example is the gravitomagnetic field due to the rotation of the Earth,\ndetected on LAGEOS data by \\cite{Ciufolini Lageos} (and which is\nalso the subject of experimental scrutiny by the Gravity Probe B and\nthe upcoming LARES missions). But when the currents seen by the observer\nvary with time --- e.g. the ones resulting from translation of the\ncelestial body, considered in \\cite{SoffelKlioner} --- then the dynamics\ndiffer significantly.\n\n\\textbf{Rotational Currents.} --- We will start by the well known\nanalogy between the electromagnetic field of a spinning charge (charge\n$Q$, magnetic moment $\\mu$) and the gravitational field (in the\nfar region $r\\rightarrow\\infty$) of a rotating celestial body (mass\n$m$, angular momentum $J$), see Fig.\\,\\ref{fig1}\n\n\\begin{figure}[h]\n\n\n\\begin{centering}\n\\includegraphics[width=123.2mm]{figv3}\n\n\\par\\end{centering}\n\n\\caption{Spinning charge vs. spinning mass }\n\n\n\\begin{centering}\n\\label{fig1} \n\\par\\end{centering}\n\\end{figure}\n\n\nThe electromagnetic field of the spinning charge is described by the\n4-potential $A^{\\alpha}=(\\phi,\\mathbf{A})$, given by (\\ref{GravPot}a).\nThe spacetime around the spinning mass is asymptotically described\nby the linearized Kerr solution, obtained by putting in (\\ref{Linear pert})\nthe perturbations (\\ref{GravPot}b) :\\begin{equation}\n\\phi=\\frac{Q}{r}\\ ,\\ \\ \\ \\mathbf{A}=\\frac{1}{c}\\frac{\\bm{\\mathbf{\\mu}}\\times\\mathbf{r}}{r^{3}}\\ \\ \\ (a);\\ \\ \\ \\ \\ \\ \\Phi=\\frac{M}{r}\\ ,\\ \\ \\ \\bm{\\mathcal{A}}=\\frac{1}{c}\\frac{\\mathbf{J}\\times\\mathbf{r}}{r^{3}}\\ ,\\ \\ \\Theta_{ij}=\\Phi\\delta_{ij}\\ \\ \\ (b).\\ \\ \\ \\label{GravPot}\\end{equation}\nFor the observer at rest $\\mathcal{O}$ the gravitational tidal tensors\nasymptotically match the electromagnetic ones, identifying the appropriate\nparameters:{\\small \\[\n\\mathbb{E}_{ij}\\simeq\\frac{M}{r^{3}}\\delta_{ij}-\\frac{3Mr_{i}r_{j}}{r^{5}}\\stackrel{M\\leftrightarrow Q}{=}E_{ij};\\ \\ \\ \\ \\mathbb{H}_{ij}\\simeq\\frac{3}{c}\\left[\\frac{(\\mathbf{r}.\\mathbf{J})}{r^{5}}\\delta_{ij}+2\\frac{r_{(i}J_{j)}}{r^{5}}-5\\frac{(\\mathbf{r}.\\mathbf{J})r_{i}r_{j}}{r^{7}}\\right]\\stackrel{J\\leftrightarrow\\mu}{=}B_{ij}\\]\n}(all the time components are zero for this observer). This me{\\small ans\nthat} $\\mathcal{O}$ will find a similarity between \\emph{physical}\n(i.e., tidal) gravitational forces and their electromagnetic counterparts:\nthe gravitational force $F_{G}^{i}=-\\mathbb{H}^{ji}S_{j}\/c$ exerted\non a gyroscope carried by $\\mathcal{O}$ is similar to the force $F_{EM}^{i}=qB^{ji}S_{j}\/2mc$\non a magnetic dipole; and the worldline deviation $D^{2}\\delta x^{i}\/d\\tau^{2}=-\\mathbb{E}^{ij}\\delta x_{i}$\nof two masses dropped from rest is similar to the deviation between\ntwo charged particles with the same $q\/m$.\n\nMoreover, observer $\\mathcal{O}$ will see test particles moving on\ngeodesics described by equations analogous to the electromagnetic\nLorentz force (see Fig. \\ref{fig1}). \n\n\\textbf{Translational Currents.} --- For the observer $\\bar{\\mathcal{O}}$\nmoving with velocity $\\mathbf{w}$ relative to the mass\/charge of\nFig. \\ref{fig1}, however, the electromagnetic and gravitational interactions\nwill look significantly different. For simplicity we will specialize\nhere to the case where $\\mathbf{J}=\\bm{\\mu}=0$, so that the mass\/charge\ncurrents seen by $\\bar{\\mathcal{O}}$ arise solely from translation.\nTo obtain the electromagnetic 4-potential $A^{\\bar{\\alpha}}$ in the\nframe $\\bar{\\mathcal{O}}$, we apply the boost $A^{\\bar{\\alpha}}=\\Lambda_{\\ \\alpha}^{\\bar{\\alpha}}A^{\\alpha}=(\\bar{\\phi},\\bar{\\mathbf{A}})$,\nwhere $\\Lambda_{\\ \\alpha}^{\\bar{\\alpha}}\\equiv\\partial\\bar{x}^{\\bar{\\alpha}}\/\\partial x^{\\alpha}$,\nusing the expansion of Lorentz transformation (as done in e.g. \\cite{WillNordvedt1972}):\n\\begin{equation}\nt=\\bar{t}\\left(1+\\frac{w^{2}}{2c^{2}}+\\frac{3w^{4}}{8c^{4}}\\right)+\\left(1+\\frac{w^{2}}{2c^{2}}\\right)\\frac{\\mathbf{\\bar{x}}.\\mathbf{w}}{c^{2}};\\ \\ \\ \\ \\mathbf{x}=\\mathbf{\\bar{x}}+\\frac{1}{2c^{2}}(\\bar{\\mathbf{x}}.\\mathbf{w})\\mathbf{w}+\\left(1+\\frac{w^{2}}{2c^{2}}\\right)\\mathbf{w}\\bar{t}\\ ,\\label{Boost}\\end{equation}\nyielding, to order $c^{-2}$, $A^{\\bar{\\alpha}}=(\\bar{\\phi},\\mathbf{\\bar{A}})$,\nwith $\\bar{\\phi}=Q(1+w^{2}\/2c^{2})\/r$ and $\\mathbf{\\bar{A}}=-Q\\mathbf{w}\/rc$.\nTo obtain $A^{\\bar{\\alpha}}$ in the coordinates ($\\bar{x}^{i},\\bar{t}$)\nof $\\bar{\\mathcal{O}}$, we must also express $r$ (which denotes\nthe distance between the source and the point of observation, in the\nframe $\\mathcal{O}$) in terms of $R\\equiv|\\mathbf{\\bar{r}}+\\mathbf{w}\\bar{t}|$,\ni.e., the distance between the source and the point of observation\nin the frame $\\bar{\\mathcal{O}}$. Using transformation (\\ref{Boost}),\nwe obtain: $r^{-1}=R^{-1}[1-(\\mathbf{w}.\\mathbf{R})^{2}\/(2R^{2}c^{2})]$,\nand finally the electromagnetic potentials seen by $\\bar{\\mathcal{O}}$:\\begin{equation}\n\\bar{\\phi}=\\frac{Q}{R}\\left(1+\\frac{w^{2}}{2c^{2}}-\\frac{(\\mathbf{w}.\\mathbf{R})^{2}}{4R^{2}c^{2}}\\right);\\ \\ \\ \\ \\ \\ \\mathbf{\\bar{A}}=-\\frac{1}{c}\\frac{Q}{R}\\mathbf{w}\\ .\\label{EMF Obar}\\end{equation}\nThe metric of the spacetime around a point mass, in the coordinates\nof $\\bar{\\mathcal{O}}$, is also obtained using transformation (\\ref{Boost}),\nwhich is accurate to Post Newtonian order, by an analogous procedure.\nFirst we apply the Lorentz boost $g_{\\bar{\\alpha}\\bar{\\beta}}=\\Lambda_{\\ \\bar{\\alpha}}^{\\alpha}\\Lambda_{\\ \\bar{\\beta}}^{\\beta}g_{\\alpha\\beta}$\nto the metric (\\ref{GravPot}) (with $\\mathcal{A}=0$); then, expressing\n$r$ in terms of $R$, we finally obtain (note that, although we are\nnot putting the bars therein, indices $\\alpha=0,i$ in the following\nexpressions refer to the coordinates of $\\bar{\\mathcal{O}}$): \\begin{eqnarray}\ng_{00} & = & -1+2\\frac{M}{Rc^{2}}+\\frac{4Mw^{2}}{Rc^{4}}-\\frac{M(\\mathbf{w}.\\mathbf{R})^{2}}{c^{4}R^{3}}\\equiv-1+\\frac{2\\bar{\\Phi}}{c^{2}};\\ \\ \\nonumber \\\\\n\\ g_{0i} & = & \\frac{4Mw_{i}}{Rc^{3}}\\equiv-\\frac{2\\bar{\\mathcal{A}}_{i}}{c^{2}};\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ g_{ij}=\\left[1+2\\frac{M}{Rc^{2}}\\right]\\delta_{ij}\\equiv\\left[1+2\\frac{\\bar{\\Theta}}{c^{2}}\\right]\\delta_{ij}\\ ,\\label{GObar}\\end{eqnarray}\nwhere we retained terms up to $c^{-4}$ in $g_{00}$, up to $c^{-3}$\nin $g_{i0}$ and $c^{-2}$ in $g_{ij}$, as usual in Post-Newtonian\napproximation. This matches, to linear order in $M$, Eqs. (5) of\n\\cite{SoffelKlioner} for the case of one single source; or e.g. Eqs.\n(11) of \\cite{Nordvedt1988} (in the case of the latter, an additional\ngauge choice, Eq. (19) of \\cite{WillNordvedt1972}, was made). The\nmetric (\\ref{GObar}), like the electromagnetic potential (\\ref{EMF Obar}),\nis now time dependent, since $\\mathbf{R}(\\bar{t})=\\mathbf{\\bar{r}}+\\mathbf{w}\\bar{t}$.\n\nThe gravitational tidal tensors seen by $\\bar{\\mathcal{O}}$ are ({\\small $\\mathbb{E}_{\\alpha0}=\\mathbb{E}_{0\\alpha}=\\mathbb{H}_{\\alpha0}=\\mathbb{H}_{0\\alpha}=0$}):{\\small \\begin{eqnarray}\n\\mathbb{E}_{ij} & = & -\\bar{\\Phi}_{,ij}-\\frac{2}{c}\\frac{\\partial}{\\partial\\bar{t}}\\bar{\\mathcal{A}}_{(i,j)}-\\frac{1}{c^{2}}\\frac{\\partial^{2}}{\\partial\\bar{t}^{2}}\\bar{\\Theta}\\delta_{ij}\\nonumber \\\\\n & = & \\frac{M\\delta_{ij}}{R^{3}}\\left[1+\\frac{3w^{2}}{c^{2}}-\\frac{9}{2}\\frac{(\\mathbf{R}.\\mathbf{w})^{2}}{c^{2}R^{2}}\\right]-\\frac{3MR_{i}R_{j}}{R^{5}}\\left[1+\\frac{2w^{2}}{c^{2}}-\\frac{5(\\mathbf{R}.\\mathbf{w})^{2}}{2c^{2}R^{2}}\\right]\\nonumber \\\\\n &  & -\\frac{3Mw_{i}w_{j}}{c^{2}R^{3}}+\\frac{6Mw_{(i}R_{j)}(\\mathbf{R}.\\mathbf{w})}{c^{2}R^{5}};\\label{Egij}\\\\\n\\mathbb{H}_{ij} & = & \\epsilon_{i}^{\\,\\,\\, lk}\\bar{\\mathcal{A}}_{k,lj}-\\frac{1}{c}\\epsilon_{ij}^{\\ \\ l}\\frac{\\partial\\bar{\\Theta}_{,l}}{\\partial\\bar{t}}\\ =\\frac{M}{cR^{3}}\\left[3\\epsilon_{ij}^{\\ \\ k}w_{k}-\\frac{3}{R^{2}}(\\mathbf{R}.\\mathbf{w})\\epsilon_{ij}^{\\ \\ k}R_{k}-\\frac{6}{R^{2}}(\\mathbf{R}\\times\\mathbf{w})_{i}R_{j}\\right],\\label{Hij}\\end{eqnarray}\n}which significantly differ from the electromagnetic ones ({\\small $E_{0\\alpha}=B_{0\\alpha}=0$}):{\\small \\begin{eqnarray}\nE_{ij} & = & -\\bar{\\phi}_{,ij}-\\frac{1}{c}\\frac{\\partial}{\\partial\\bar{t}}\\bar{A}_{i;j}\\ =\\ E_{i,j}\\nonumber \\\\\n & = & \\frac{Q\\delta_{ij}}{R^{3}}\\left[1+\\frac{w^{2}}{2c^{2}}-\\frac{3}{4}\\frac{(\\mathbf{R}.\\mathbf{w})^{2}}{c^{2}R^{2}}\\right]-\\frac{3QR_{i}R_{j}}{R^{5}}\\left[1+\\frac{w^{2}}{2c^{2}}-\\frac{5(\\mathbf{R}.\\mathbf{w})^{2}}{4c^{2}R^{2}}\\right]\\nonumber \\\\\n &  & -\\frac{Qw_{i}w_{j}}{2c^{2}R^{3}}+\\frac{3Qw_{[i}R_{j]}(\\mathbf{R}.\\mathbf{w})}{c^{2}R^{5}};\\label{Eij}\\\\\nE_{i0} & = & -\\frac{1}{c}\\frac{\\partial}{\\partial\\bar{t}}\\bar{\\phi}_{;i}-\\frac{1}{c^{2}}\\frac{\\partial^{2}\\bar{A}_{i}}{\\partial\\bar{t}^{2}}\\ \\equiv\\frac{1}{c}\\ \\frac{\\partial E_{i}}{\\partial\\bar{t}}\\ =\\ \\frac{Q}{cR^{3}}\\left[w_{i}-\\frac{3(\\mathbf{R}.\\mathbf{w})R_{i}}{R^{2}}\\right];\\label{Ei0}\\\\\nB_{ij} & = & \\epsilon_{i}^{\\ lm}\\bar{A}_{m;lj}\\ \\equiv\\ B_{i,j}=\\frac{Q}{cR^{3}}\\left[\\epsilon_{ij}^{\\ \\ k}w_{k}-\\frac{3}{R^{2}}(\\mathbf{R}\\times\\mathbf{w})_{i}R_{j}\\right];\\label{Bij}\\\\\nB_{i0}\\  & = & \\frac{1}{c}\\frac{\\partial B_{i}}{\\partial\\bar{t}}=-\\ \\frac{3Q}{c^{2}R^{5}}(\\mathbf{R}.\\mathbf{w})(\\mathbf{R}\\times\\mathbf{w})_{i}\\ .\\label{Bi0}\\end{eqnarray}\n}Note in particular that, unlike their gravitational counterparts,\n$E_{\\alpha\\beta}$ and $B_{\\alpha\\beta}$ are not symmetric, and have\nnon-zero time components. The antisymmetric parts $E_{[ij]}=E_{[i,j]}$\nand $B_{[ij]}=B_{[i,j]}$ above are (vacuum) Maxwell equations $\\nabla\\times\\mathbf{E}=-(1\/c)\\partial\\mathbf{B}\/\\partial t$\nand $\\nabla\\times\\mathbf{B}=(1\/c)\\partial\\mathbf{E}\/\\partial t$,\nimplying that a time varying electric\/magnetic field endows the magnetic\/electric\ntidal tensor with an antisymmetric part. For instance, a time varying\nelectric field will always induce a force on a magnetic dipole. The\nfact that $\\mathbb{E}_{\\alpha\\beta}$ and $\\mathbb{H}_{\\alpha\\beta}$\nare symmetric reflects the absence of analogous gravitational effects.\nThe time component $B_{i0}$ means that the force on a magnetic dipole\n(magnetic moment $\\mu=q\/2m$) will have a time component $(F_{EM})_{0}=(1\/c)\\bm{\\mu}.\\partial\\mathbf{B}\/\\partial t$,\nwhich (see \\cite{CostaHerdeiro2009} sec. 1.2) is minus the power\ntransferred to the dipole by Faraday's law of induction (and is reflected\nin the variation of the dipole's proper mass $m=-P^{\\alpha}U_{\\alpha}\/c^{2}$).\nAgain, this is an effect which has no gravitational counterpart: $\\mathbb{H}_{\\alpha0}=\\mathbb{H}_{0\\alpha}=0$,\nthus $(F_{G})_{0}=0$, and the proper mass of the gyroscope is a constant\nof the motion. \n\nThe space part of the geodesic equation for a test particle of velocity\n$\\mathbf{v}$ is:{\\small \\begin{eqnarray}\n\\mathbf{a} & = & \\nabla\\bar{\\Phi}+\\frac{2}{c}\\frac{\\partial\\bm{\\bar{\\mathcal{A}}}}{\\partial\\bar{t}}-2\\mathbf{v}\\times(\\nabla\\times\\bm{\\bar{\\mathcal{A}}})-\\frac{3}{c^{2}}\\frac{\\partial}{\\partial\\bar{t}}\\left(\\frac{M}{R}\\right)\\mathbf{v}\\label{GeoTrans}\\\\\n & = & -\\frac{M}{R^{3}}\\left[1+\\frac{2w^{2}}{c^{2}}-\\frac{3(\\mathbf{R}.\\mathbf{w})^{2}}{2c^{2}R^{2}}\\right]\\mathbf{R}+\\frac{3M(\\mathbf{R}.\\mathbf{w})}{c^{2}R^{3}}\\mathbf{w}-\\frac{4M}{c^{2}R^{3}}\\mathbf{v}\\times(\\mathbf{R}\\times\\mathbf{w})+\\frac{3}{c^{2}}\\frac{M}{R^{3}}(\\mathbf{R}.\\mathbf{w})\\mathbf{v}\\ ,\\nonumber \\end{eqnarray}\n}which matches equation (10) of \\cite{SoffelKlioner}, or (7) of \\cite{Nordvedt1973},\nagain, in the special case of only one source, and keeping therein\nonly linear terms in the perturbations and test particle's velocity\n$\\mathbf{v}$. \n\nComparing with its electromagnetic counterpart{\\small \\[\n\\left(\\frac{m}{q}\\right)\\mathbf{a}=\\mathbf{E}+\\frac{\\mathbf{v}}{c}\\times\\mathbf{B}=\\frac{Q}{R^{3}}\\left[1+\\frac{w^{2}}{2c^{2}}-\\frac{3(\\mathbf{R}.\\mathbf{w})^{2}}{4c^{2}R^{2}}\\right]\\mathbf{R}-\\frac{1}{2}\\frac{Q(\\mathbf{R}.\\mathbf{w})}{c^{2}R^{3}}\\mathbf{w}+\\frac{Q}{c^{2}R^{3}}\\mathbf{v}\\times(\\mathbf{R}\\times\\mathbf{w})\\]\n}we find them similar to a certain degree (up to some factors), except\nfor the last term of (\\ref{GeoTrans}). That term signals a difference\nbetween the two interactions, because it means that there is a velocity\ndependent acceleration which is parallel to the velocity; that is\nin contrast with the situation in electromagnetism, where the velocity\ndependent accelerations arise from magnetic forces, and are thus always\nperpendicular to $\\mathbf{v}$.\n\nAs expected from Eqs. (\\ref{PrecessGen}) (and by contrast with the\nother effects), the precession of a gyroscope carried by $\\bar{\\mathcal{O}}$,\nEq. (\\ref{PrecessTrans}b) takes a form analogous to the precession\nof a magnetic dipole, Eq. (\\ref{PrecessTrans}a), if we express $\\mathbf{S}$\nin the local orthonormal tetrad $e^{\\hat{i}}$, non rotating relative\nto the inertial observer at infinity, such that $S^{i}=(1-M\/R)S^{\\hat{i}}$:\n\\begin{equation}\n\\frac{d\\mathbf{S}}{d\\bar{t}}=\\frac{q}{2m}\\frac{Q}{c^{2}R^{3}}\\left[\\mathbf{S}\\times(\\mathbf{R}\\times\\mathbf{w})\\right]\\ \\ (a);\\ \\ \\ \\ \\ \\ \\frac{dS^{\\hat{i}}}{d\\bar{t}}=\\frac{2M}{c^{2}R^{3}}\\left[(\\mathbf{R}\\times\\mathbf{w})\\times\\mathbf{S}\\right]^{\\hat{i}}\\ \\ (b)\\ .\\label{PrecessTrans}\\end{equation}\nIf instead of the gyroscope comoving with observer $\\bar{\\mathcal{O}}$\n(with constant velocity $\\mathbf{w}$), we had considered a gyroscope\nmoving in a circular orbit, then an additional term would arise in\nanalogy with Thomas precession for the magnetic dipole; for a circular\ngeodesic that term amounts to $-1\/4$ of expression (\\ref{PrecessTrans}b),\nand we would obtain the well known equation for geodetic precession\n(e.g. \\cite{Gravitation and Inertia}).\n\n\n\\section{Conclusion}\n\nWe conclude our paper by discussing some of the implications of our\nconclusions in the approaches usually found in literature. In the\nframework of linearized theory, e.g. \\cite{Ruggiero:2002,Gravitation and Inertia},\nEinstein equations are often written in a Maxwell-like form; likewise,\ngeodesics, precession and gravitational force on a spinning test particle\nare cast (in terms of 3-vectors defined in analogy with the electromagnetic\nfields $\\mathbf{E}$ and $\\mathbf{B}$) in a form similar to, respectively,\nthe Lorentz force on a charged particle, the precession and the force\non a magnetic dipole. \n\nWe have concluded that the actual physical similarities between gravity\nand electromagnetism (on which the physical content of such approaches\nrelies) occur only on very special conditions. For tidal effects,\nlike the forces on a gyroscopes\/dipoles, the analogy manifest in Eqs.\n(\\ref{FG_Stationary}) holds only when the \\emph{test particle} sees\ntime independent \\emph{fields}. In the example of analogous systems\nconsidered in section \\ref{Translational-vs.-Rotational}, this means\nthat the center of mass of the gyroscope\/dipole must not move relative\nto the central body. In the case of the analogy between the equation\nof geodesics and the Lorentz force law (see Fig. \\ref{fig1}), as\nmanifest in equation (\\ref{geoGeneral}), it is in the \\emph{potentials\/metric\nperturbations}, as seen by \\emph{the observer} (not the test particle!),\nthat the time independence is required. The latter condition is not\nas restrictive as the one of the tidal effects: consider for instance\nobservers moving in circular orbits around a static mass\/charge; such\nobservers see an unchanging spacetime, and unchanging electromagnetic\npotentials, so, for them, the equation of geodesics and Lorentz force\ntake similar forms (such analogy may actually be cast in an exact\nform, see \\cite{Natario,Jantzen}). However, those observers see a\ntime-varying electric field $\\mathbf{E}$ (constant in magnitude,\nbut varying in direction), which, by means of equations (4) and (6),\nimplies that the tidal tensors are not similar to the gravitational\nones%\n\\footnote{The electromagnetic field $F^{\\alpha\\beta}$ is not constant along\nthe worldline of an observer moving in a circular orbit (radius $R$,\nangular velocity $\\bm{\\Omega}$, velocity $\\mathbf{w}=\\bm{\\Omega}\\times\\mathbf{R}$)\naround a point charge. Its variation endows the magnetic tidal tensor\nwith an antisymmetric part, and the electric tidal tensor with a time\ncomponent: $dF^{0i}\/d\\tau=Qw^{i}\/cR^{3}=-2E^{[i0]}=-\\epsilon^{ijk}B_{[jk]}$.\nThis means that they significantly differ from the GR tidal tensors\nseen by an observer in circular motion around a point mass.\n\nNote that both the GR and the EM tidal tensors for these analogous\nproblems can be obtained from, respectively, Eqs. (\\ref{Egij})-(\\ref{Hij})\nand (\\ref{Eij})-(\\ref{Bi0}), making therein $\\mathbf{R}.\\mathbf{w}=0$\n(corresponding to circular motion), despite the fact that these expressions\nwere originally derived for an observer with constant velocity. This\nis because, as can be seen from their definitions in Table \\ref{analogy},\nit is the 4-velocity $U^{\\alpha}$ (regardless of the way it varies),\nat the given point, that determines the tidal tensors. %\n}. \n\nFinally, as a consequence of this analysis, a distinction, from the\npoint of view of the analogy with electrodynamics, between effects\nrelated to (stationary) rotational mass currents, and those arising\nfrom translational mass currents, becomes clear: albeit in the literature\nboth are dubbed {}``gravitomagnetism'', one must note that, while\nthe former are clearly analogous to magnetism, in the case of the\nlatter the analogy is not so close.\n\n\\vskip 20pt\n\\leftline{{\\normalfont\\bfseries Acknowledgments}}\n\\vskip 4pt\nWe thank the anonymous referee for very useful comments and suggestions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\nDiluted magnetic semiconductors (DMS) combine the benefits of semiconducting and magnetic materials and hence are most promising candidates for future spintronics applications \\cite{Dietl2014, Dietl2010}. In particular, III-Mn-V DMS, as for instance the archetypal compound Ga$_{1-x}$Mn$_{x}$As, have been the subject of intensive research in the last two decades, motivated by both fundamental and technological interests \\cite{Ohno1998}.  \nIn Ga$_{1-x}$Mn$_{x}$As, as well as in the related compound Ga$_{1-x}$Mn$_{x}$P, a few atomic percent of the nonmagnetic gallium host sublattice atoms are replaced with magnetic manganese atoms acting as acceptors. Both materials exhibit a long-range magnetic order of substitutional Mn ions mediated by holes \\cite{Jungwirth2006, Jungwirth2007, Scarpulla2005, Winkler2011}. Crystalline defects---in particular Mn interstitials (Mn\\textsubscript{I}), which act as double donors and thereby compensate the hole doping, as well as As antisites (As\\textsubscript{Ga})---play a substantial role for both the strongly interrelated electronic and magnetic properties of DMS. Therefore, the preparation of high-quality samples is a delicate procedure and constitutes an obstacle to further enhancement of the Curie temperature $T_\\mathrm{C}$ beyond $190\\,$K for Ga$_{1-x}$Mn$_{x}$As \\cite{Chen2009} and $65\\,$K for Ga$_{1-x}$Mn$_{x}$P \\cite{Farshchi2006}. \nMoreover, the effects of disorder \\cite{Berciu2001} and carrier-carrier interactions \\cite{Richardella2010} make the theoretical description difficult \\cite{Sato2010}, and there is still no consensus on the development of spontaneous magnetization in these materials.  \nIn literature, a broad spectrum of sometimes opposing theoretical approaches exists \\cite{Samarth2012}, ranging from the assumption of free charge carriers \\cite{Koenig2003, Dietl2001, Dietl2000, Koenig2000} to the opposite case of strongly localized carriers \\cite{Dobrowolska2012, Bhatt2002, Berciu2001}. While the first picture assumes the Fermi energy to lie within a merged band consisting of the valence band and a strongly broadened impurity band, the second model proposes the existence of a separate impurity band.\nThe latter, so-called impurity-band model is expected to be valid for weakly-doped samples. In this context, Kaminski and Das Sarma \\cite{Kaminski2002, Kaminski2003} have developed an analytic polaron percolation theory for DMS ferromagnetism in the limit of strong charge carrier localization and for an inhomogeneous spatial distribution of magnetic impurities. In this model, the spins of localized charge carriers can polarize the surrounding magnetic impurities, leading to the emergence of bound magnetic polarons (BMP), which grow in size for decreasing temperatures and at low enough temperatures overlap until an infinite cluster is formed and spontaneous magnetization occurs \\cite{Kumar2013}.\nAnother conceivable scenario is that the p-d Zener model, which is equivalent to a weak-coupling RKKY picture and believed to be appropriate for more metallic systems with free charge carriers, can also be applied on the insulator side of the metal-insulator transition (MIT) of Ga$_{1-x}$Mn$_{x}$As. In this case, the hole localization length remains much greater than the average distance between the acceptors \\cite{Ohno2008, Dietl2006}. Large mesoscopic fluctuations in the local value of the density of states near the MIT are expected to lead to a nano-scale phase separation into ferromagnetic and paramagnetic regions below $T_\\mathrm{C}$, whereby the paramagnetic (hole-poor) regions can persist down to low temperatures, coexisting with ferromagnetic (hole-rich) bubbles. \\newline\nMotivated by the still ongoing debate about the particular mechanisms of ferromagnetism and the crucial role of point defects, the present study focuses on a systematic investigation of low-frequency electronic transport properties on a series of Ga$_{1-x}$Mn$_{x}$As samples with different Mn contents and growth parameters. We apply fluctuation spectroscopy, a method which is sensitive to electric inhomogeneities on the nano- and micrometer scale and has been proven to be a versatile tool for identifying electronic phase separation and magnetically driven percolation, as observed, e.g., for perovskite manganites \\cite{Podzorov2000} and more recently for the semimetallic ferromagnet EuB\\textsubscript{6} \\cite{Das2012}. Here, we aim to compare the noise behavior of metallic Ga$_{1-x}$Mn$_{x}$As films ($x>2$ \\%) with rather insulating ($x<2$ \\%) samples in order to gain further insight into possible percolative transitions and electronic phase separation. Moreover, we investigate another compound, Ga$_{1-x}$Mn$_{x}$P, an even better candidate for the observation of a percolative transition. Despite its chemical similarity with Ga$_{1-x}$Mn$_{x}$As, Ga$_{1-x}$Mn$_{x}$P has a Mn acceptor level lying four times deeper within the band gap, i.e.\\ about $0.4\\,$eV above the valence band edge \\cite{Clerjaud1985}. Therefore, it is obvious that the charge carriers are of a much more localized nature than in Ga$_{1-x}$Mn$_{x}$As. Nevertheless, hole-mediated ferromagnetism has also been demonstrated in Ga$_{1-x}$Mn$_{x}$P \\cite{Farshchi2006}. In addition, Ga$_{1-x}$Mn$_{x}$P is very similar to Ga$_{1-x}$Mn$_{x}$As concerning the magnetic anisotropy, spin-polarization and the scaling of $T_\\mathrm{C}$ as a function of the Mn concentration \\cite{Zhou2015}. A comparison of weakly doped Ga$_{1-x}$Mn$_{x}$As with a Ga$_{1-x}$Mn$_{x}$P sample ($x=3.5$ \\%) may provide valuable information about possible commonalities and differences with regard to their noise characteristics and possible electronic inhomogeneities. Apart from varying the Mn content of the above mentioned compounds, an alternative approach for the control of magnetic and electronic properties is the irradiation with He ions, leading to the introduction of deep traps into the system and thereby to an increasing disorder. In this work, we therefore also address the question of which consequences ion irradiation has for the low-frequency resistance noise characteristics and whether signatures of an electronic phase separation or introduced defects can be observed for irradiated samples. It should be noted that fluctuation spectroscopy in general is highly suitable for studying the energy landscape of defects in semiconducting thin films \\cite{Lonsky2016, Lonsky2015} and semiconductor heterostructures \\cite{Mueller2006, Mueller2006a, Muller2015}.      \n\n\\section{Experimental Details}\n\\subsection{Sample Growth} \\label{growth}\nElectronic transport measurements have been performed on a total of seven Ga$_{1-x}$Mn$_{x}$As thin film samples and one Ga$_{1-x}$Mn$_{x}$P sample, see Table \\ref{tab:table1} for an overview. For all films, the corresponding Curie temperatures were determined by magnetization measurements \\cite{Zhou2016, Hamida2014, Yuan2017a, Zhou2015}.\nIn general, a crucial parameter is the Mn content $x$, but other factors as thermal annealing or induced disorder by ion irradiation also have a strong influence on the Curie temperature $T_\\mathrm{C}$ and the crystalline defect characteristics \\cite{Hayashi2001, Potashnik2001, Esch1997, Sinnecker2010}.\nThe samples in this study were prepared in two different ways. Metallic Ga$_{1-x}$Mn$_{x}$As samples with $x=4$ \\% were grown by low-temperature molecular beam epitaxy (LT-MBE) on semi-insulating GaAs(001) substrates in a Mod Gen II MBE system with the lowest possible As\\textsubscript{2}-partial pressure of about $2\\times 10^{-6}\\,$mbar at PTB in Braunschweig \\cite{Hamida2014}. After the growth of a $100\\,$nm high-temperature (HT) GaAs buffer layer at $T_\\mathrm{g}=560\\,^\\circ$C, the temperature was decreased to $270\\,^\\circ$C for the subsequent LT Ga$_{1-x}$Mn$_{x}$As growth. Post-growth annealing at $200\\,^\\circ$C (18 hours in ambient atmosphere) was performed for one of the samples in order to enhance $T_\\mathrm{C}$. The total Mn concentration $x$ was calculated from the molecular flux ratio of Mn and Ga measured in the MBE at the position of the wafer and compared with reflection high-energy electron diffraction (RHEED) and energy-dispersive X-ray spectroscopy (EDX) measurements.   \nMoreover, three Ga$_{1-x}$Mn$_{x}$As samples with a nominal Mn concentration of $x=6$ \\% were grown by LT-MBE on semi-insulating GaAs(001) using a Veeco Mod III MBE system in Nottingham \\cite{Zhou2016, Wang2008}. In this case, thermal annealing was performed at $190\\,^\\circ$C for 48 hours in ambient atmosphere and the Mn content was determined from the Mn\/Ga flux ratio. Two of the films were irradiated with different doses of He ions after growth. This particular method allows to control the hole concentration and thus the electronic as well as the magnetic properties without changing the Mn content of a sample. The He-ion energy was chosen as $4\\,$keV. The fluences were $2.5\\times 10^{13}\\,$cm$^{-2}$ and $3.5\\times 10^{13}\\,$cm$^{-2}$ for the two irradiated samples. A better measure for the effect of irradiation on material properties than the fluence is the so-called displacement per atom (DPA), i.e.\\ the number of times that an atom in the target is displaced during irradiation. This allows for a comparison with data reported in the literature, in which other ion species and energies are used. For the two irradiated samples, the DPA was $1.6\\times 10^{-3}$ and $2.24\\times 10^{-3}$, respectively \\cite{Zhou2016}. During ion irradiation, the films were tilted by $7^{\\circ}$ to avoid channeling. The irradiation parameters result in defects distributed roughly uniformly in the whole Ga$_{1-x}$Mn$_{x}$As layer as confirmed by simulations using the SRIM (Stopping and Range of Ions in Matter) code \\cite{Ziegler1985}. No measurable increase of Mn interstitials was observed by Rutherford backscattering spectroscopy (RBS) \\cite{Zhou2016}. Previous studies show that also the sheet concentration of substitutional Mn atoms remains constant \\cite{Winkler2011}, which is why we conclude that the main effect of He-ion irradiation is to introduce deep traps and thereby compensate the holes. It is well established that these defects reside in the As sublattice and most of them are primary defects related to vacancies and interstitials \\cite{Sinnecker2010, Pons1985}. In our case, atomic force microscopy (AFM) and RBS measurements do not show any indications of irradiation induced surface reconstruction. Further details can be found in Ref.\\ \\cite{Zhou2016}.    \nFinally, two Ga$_{1-x}$Mn$_{x}$As samples with low Mn contents of $1.8$ \\% and $1.2$ \\% as well as a Ga$_{1-x}$Mn$_{x}$P sample with $x=3.5$ \\% were fabricated by ion implantation combined with pulsed laser melting in Dresden \\cite{Yuan2017a, Zhou2015}. Ion implantation is a common materials engineering technique for introducing foreign ions into a host material. In this case, Mn ions are implanted into GaAs or GaP wafers. The subsequent laser pulse drives a rapid liquid-phase epitaxial growth. The implantation energy was set to $100\\,$keV for GaAs \\cite{Yuan2017a} and $50\\,$keV for GaP \\cite{Yuan2016, Zhou2015}. The wafer normal was tilted by $7^{\\circ}$ with respect to the ion beam to avoid a channeling effect. A coherent XeCl laser (with $308\\,$nm wavelength and $28\\,$ns duration) was employed to recrystallize the samples, and the energy densities were optimized to achieve high crystalline quality and the highest Curie temperature. The optimal laser energy density is $0.30\\,$J\/cm$^2$ for Ga$_{1-x}$Mn$_{x}$As and $0.45\\,$J\/cm$^2$ for Ga$_{1-x}$Mn$_{x}$P. The Mn concentration was determined by secondary ion mass spectroscopy. In contrast to films grown by LT-MBE, neither Mn interstitials nor As antisites are observed in samples prepared by ion implantation combined with pulsed laser melting \\cite{Yuan2017a, Cho2007}. Transmission electron microscopy (TEM) studies prove the high crystalline quality of the films and exclude the presence of any extended lattice defects, amorphous inclusions and precipitates of other crystalline phases \\cite{Yuan2017a}.     \nFor some selected films, an array of $50\\times 50\\, \\mu$m$^2$ Hall bars was defined by photolithography followed by wet chemical etching. The quality of Hall effect measurements thereby improves significantly due to a well-defined contact geometry and, as explained in more detail below, the resistance noise magnitude as the desired measurement signal increases due to smaller sample volumes according to Hooge's law \\cite{Hooge1969, Hooge1976}. For all samples, electrical contacts were made by soldering In\/Sn on top of the films.\nCharge carrier concentrations obtained from Hall effect measurements are given in Table \\ref{tab:table1}. Due to the use of two different fabrication techniques and since the LT-MBE samples from Braunschweig and Nottingham were grown at different substrate temperatures and As-fluxes, care has to be taken in comparing the values for the hole concentration $p$ of films of different origin. Apart from that, as expected, the hole density increases after thermal annealing for the $x=4$ \\% samples and decreases with increasing ion irradiation dose for the $x=6$ \\% samples, cf.\\ Section \\ref{ssecmetallic} for more details. Furthermore, there is also a clear correlation between $T_\\mathrm{C}$ and $p$. Finally, we point out that extensive studies on all present samples, including magnetization measurements and standard thin film characterization techniques, have been published elsewhere, cf.\\ Ref.\\ \\cite{Hamida2014, Zhou2016, Yuan2017a, Zhou2015}.      \n\\begin{table*}\n\\caption{\\label{tab:table1}Overview of the investigated thin film samples and related parameters, including: information about the manganese content $x$, whether samples were grown by low-temperature molecular beam epitaxy (LT-MBE) or ion implantation combined with pulsed laser melting (II+PLM), the institute where samples have been fabricated, which kind of post-treatment was given, the values of the film thickness, the Curie temperature $T_\\mathrm{C}$ as determined by magnetization measurements, and the hole density $p(T=300\\,\\mathrm{K})$ obtained from Hall effect measurements.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccc}\nMn content&Fabrication&Source&Remarks&Thickness&$T_\\mathrm{C}$ & $p(T=300\\,\\mathrm{K})\\ [1\/\\mathrm{cm}^3]$\\\\\n\\hline\nGa$_{1-x}$Mn$_{x}$As\\\\\n4.0 \\% & LT-MBE & PTB & as-grown, micro-structured& $25\\,$nm & $70\\,$K & $8.0\\times 10^{19}$\\\\\n4.0 \\% & LT-MBE & PTB & annealed ($200\\,^\\circ$C, $18\\,$h), micro-structured& $25\\,$nm & $110\\,$K & $1.2\\times 10^{20}$\\\\\n6.0 \\% & LT-MBE & Nottingham & annealed ($190\\,^\\circ$C, $48\\,$h)& $25\\,$nm & $125\\,$K & $9.6\\times 10^{20}$\\\\\n6.0 \\% & LT-MBE & Nottingham & annealed, He-ion irradiated (low dose)& $25\\,$nm & $75\\,$K & $8.0\\times 10^{20}$\\\\\n6.0 \\% & LT-MBE & Nottingham & annealed, He-ion irradiated (high dose)& $25\\,$nm & $50\\,$K & $5.6\\times 10^{20}$\\\\\n1.8 \\% & II+PLM & HZDR & as-grown, micro-structured& $60\\,$nm & $60\\,$K & $2.8\\times 10^{20}$\\\\\n1.2 \\% & II+PLM & HZDR & as-grown, micro-structured& $60\\,$nm & $31\\,$K & $1.0\\times 10^{20}$\\\\\n\\hline\nGa$_{1-x}$Mn$_{x}$P\\\\\n3.5 \\% & II+PLM & HZDR & as-grown& $34\\,$nm & $45\\,$K & $3.1\\times 10^{20}$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\\subsection{Measurements}\nElectronic transport measurements have been performed using both AC and DC techniques. Experiments were carried out in a continuous-flow cryostat with variable temperature insert. Magnetic fields were applied perpendicular to the film plane.\nFor some of the lithographically patterned Ga$_{1-x}$Mn$_{x}$As samples, low-frequency noise spectroscopy was conducted in a five-terminal AC setup, where the sample is placed in a bridge-circuit in order to suppress the constant DC voltage offset and to minimize external perturbations \\cite{Scofield1987}. Other samples were measured in a four-terminal AC or DC setup. As a few samples showed a frequency-dependent resistivity at low temperatures, the excitation frequency was reduced from $227\\,$Hz down to $17\\,$Hz, or a DC noise measurement setup was utilized, which was verified to yield the same results as AC noise measurements.        \nThe fluctuating voltage signal is preamplified and processed by a spectrum analyzer yielding the voltage noise power spectral density (PSD) $S_{V}(\\omega)$ defined by:\n\\begin{equation}\nS_{V}(\\omega)=2\\lim\\limits_{T \\to \\infty}\\frac{1}{T}\\left| \\int_{-T\/2}^{T\/2} {\\rm d}t \\, e^{-i\\omega t} \\, \\delta V(t) \\right|^2,\n\\end{equation}        \nwhere $\\delta V(t)$ represents the fluctuating voltage drop across the sample and $\\omega = 2\\pi f$ the angular frequency. \nCare was taken that all spurious sources of noise were minimized or eliminated. \nAs required by Hooge's empirical law \\cite{Hooge1969, Hooge1976},\n\\begin{equation}\nS_{V}(f)=\\frac{\\gamma_{H}\\cdot V^2}{n \\Omega f^{\\alpha}},\n\\end{equation}   \nthe magnitude of the voltage noise scales as $S_V \\propto V^2 \\propto I^2$, where $I$ represents the current flowing through the sample. \nHere, $n$ is the charge carrier density and $\\Omega$ the 'noisy' sample volume, i.e.\\ $n \\Omega = N_c$ gives the total number of  charge carriers in the material causing the observed $1\/f$ noise. $\\alpha$ describes the frequency exponent which is commonly in the range $0.8\\leq \\alpha \\leq 1.4$ for $1\/f$-type fluctuations.  \nThe Hooge parameter $\\gamma_H$ is widely used to compare the noise level of different systems and covers a range of $\\gamma_H=10^{-6}$--$10^{7}$ for different materials \\cite{Raquet2001}. For bulk semiconductors, $\\gamma_H$ usually is of order $10^{-2}$--$10^{-3}$. Moreover, it is useful to normalize the magnitude of the voltage fluctuations with respect to the applied current, $S_{R}=S_{V}\/I^2$, and to the resistance of the sample, resulting in $S_{R}\/R^2$.\nExemplary noise spectra for a Ga$_{1-x}$Mn$_{x}$As sample are presented in Fig.$\\,$\\ref{PTB_CURRENT_DEP} for three different currents in a log-log plot. The dashed line represents a $1\/f^{\\alpha}$ function with a slope of $\\alpha =1.2$. The inset clarifies the quadratic scaling of $S_{V}$ with the current $I$ as predicted by Hooge's law. \nFurther details about the fluctuation spectroscopy technique can be found in \\cite{Raquet2001,Mueller2011}.       \n\\begin{figure\n\\centering\n\\includegraphics[width=8.5 cm]{PTB_1019_8_currentdep.eps}\n\\caption{Exemplary current-dependent noise spectra $S_{V}(f)$ acquired for the as-grown Ga$_{1-x}$Mn$_{x}$As sample with $x=4$ \\%. Inset: A quadratic dependence (red line: linear fit to the data) between $S_{V}$ and $I$ verifying Hooge's empirical law.} \n\\label{PTB_CURRENT_DEP}%\n\\end{figure}%\n\n\\section{Results and Discussion}\n\n\\subsection{Ga\\textsubscript{1-x}Mn\\textsubscript{x}As films with high Mn content} \\label{ssecmetallic}\n\nAt first, we will discuss measurements on metallic films with Mn contents $x>2$ \\%. Resistivity curves of as-grown and annealed Ga$_{1-x}$Mn$_{x}$As samples with $x=4$ \\% are depicted in Fig.$\\,$\\ref{PTB_PSD}(a). As expected, typical maxima \\cite{Matsukura1998} are observed in the vicinity of the samples' respective Curie temperature $T_\\mathrm{C}$. After post-growth annealing, the resistivity significantly decreases while $T_\\mathrm{C}$ increases compared to as-grown films. This effect can be explained by the out-diffusion of Mn interstitials to the film surface \\cite{Edmonds2004}. Commonly, these interstitials act as donors and compensate the hole-mediated ferromagnetism. \nFig.$\\,$\\ref{PTB_PSD}(b) shows the corresponding normalized magnitude of the resistance fluctuations $S_{R}\/R^2$ at $1\\,$Hz as a function of temperature in a semi-logarithmic representation. Remarkably, the PSD varies over several orders of magnitude for the two films and shows, in contrast to the resistivity, no significant features around $T_\\mathrm{C}$. More specifically, there are no major changes in the noise behavior throughout the entire temperature range. In addition, a constant external out-of-plane magnetic field up to $7\\,$T does not lead to any changes in the normalized resistance noise (not shown, cf.\\ Ref.\\ \\cite{Lonsky2017}). Possible contributions from mixed phases could be overshadowed by fluctuations related to thermally-activated impurity switching processes. However, more likely, due to the high concentration of Mn substitutional atoms, charge carriers are delocalized and the formation of magnetic polarons is not to be expected for these metallic samples. The observed $1\/f$-type noise is likely to be dominated by switching processes related to crystalline defects, which can also be seen in a strong variation of the Hooge parameter $\\gamma_H$ for the two different samples. At room temperature, we obtain $\\gamma_H=1\\times 10^{-2}$ for the as-grown and $\\gamma_H=3\\times 10^0$ for the annealed film. Apparently, thermal annealing has a strong influence on the noise magnitude, leading to the presumption that slow fluctuation processes related to Mn interstitials might play an important role. Since thermal annealing reduces the density of Mn interstitials, one might expect a lower PSD for annealed samples, but the opposite is the case. The same behavior is observed for two $x=7$ \\% samples with a thickness of $d=25\\,$nm (not shown). Strikingly, for two further $x=4$ \\% films with a higher thickness of $d=100\\,$nm and similar values of $T_{\\mathrm{C}}$, a slight decrease of the noise magnitude is observed after thermal annealing, indicating that the rearrangement of Mn interstitials due to diffusion processes towards the surface \\cite{Edmonds2004}, which become passivated due to oxidation or by binding with surplus As atoms, and concomitant surface effects may play an important role for the changes in the PSD after thermal annealing.\n   \n\\begin{figure\n\\centering\n\\includegraphics[width=8.5 cm]{PTB_Noise_Res_4.eps}\n\\caption{(a) Temperature-dependent resistivity showing characteristic maxima around $T_\\mathrm{C}$ (marked by arrows) and (b) normalized resistance noise magnitude of as-grown and annealed Ga$_{1-x}$Mn$_{x}$As samples with 4 \\% Mn content. No features in the noise power are visible around $T_\\mathrm{C}$.} \n\\label{PTB_PSD}%\n\\end{figure}%\nIn order to deduce the characteristic energies of the switching processes contributing to the $1\/f$-type noise, we apply the phenomenological model of Dutta, Dimon and Horn (DDH) \\cite{Dutta1979}. In this model, a certain distribution of activation energies $D(E,T)$ determines the temperature dependence of both the noise magnitude and the frequency exponent. An essential requirement for the applicability of this model is to check whether $\\alpha(T)$ calculated after\n\\begin{equation} \\label{eq:DDH}\n\\alpha(T)=1-\\frac{1}{\\ln(2\\pi f\\tau _0)} \\left[\\frac{\\partial \\ln(\\frac{S_R}{R^2}(f,T))}{\\partial \\ln(T)}-g^{\\prime}(T)-1\\right]\n\\end{equation}   \nagrees with the measured values. Here, $\\tau _0$ represents an attempt time, usually between $10^{-14}$ and $10^{-11}\\,$s, corresponding to typical inverse phonon frequencies. Moreover, it is $g^{\\prime}(T)=\\frac{\\partial \\ln(g(T))}{\\partial \\ln(T)} \\equiv b$, where $g(T)=a \\cdot T^{b}$ accounts for an explicit temperature dependence of the distribution of activation energies $D(E,T)$, which can be caused by a change of the number of thermally activated switching events (excitation of defect states) or of the coupling of fluctuations to the resistivity with temperature. As can be seen in Fig.$\\,$\\ref{PTB_DDH}(a) and (b), we find a good qualitative or even quantitative agreement between model and experiment for both $x=4$ \\% samples. This allows for calculating the distribution of activation energies,\n\\begin{equation}\nD(E) \\propto \\frac{2\\pi f}{k_B T}\\frac{1}{g(T)}\\frac{S_R}{R^2}(f,T), \n\\end{equation}\nof thermally activated fluctuators, which is shown in Fig.$\\,$\\ref{PTB_DDH}(c) and (d). Here, it is $E=-k_B T \\ln(2\\pi f \\tau_{0})$. Both samples show a similar behavior, namely an increase of $D(E)$ towards higher activation energies, which we interpret as a superposition of several thermally activated processes with different energies, that can be attributed to various kinds of defects. In both cases, four local maxima in $D(E)$ are observed and marked by black arrows in Fig.$\\,$\\ref{PTB_DDH}(c) and (d), whereby the values of the corresponding activation energies are remarkably similar for the two samples. It is plausible to assume that, due to the low growth temperatures utilized during LT-MBE, which are required in order to prevent phase separation within the material, a great variety of defects, such as Mn interstitials and As antisites, contribute to the distribution of activation energies. Although the energies of the local maxima are comparable to typical impurity binding energies in GaAs, care has to be taken when assigning the energies to specific defect states due to band gap renormalization in heavily doped semiconductors, which is accompanied by a shift of the respective binding energies.       \n\\begin{figure\n\\centering\n\\includegraphics[width=8.5 cm]{PTB_samples_DDH_DE_4percent.eps}\n\\caption{(a) and (b) Application of the phenomenological DDH model (red curves) to the resistance noise data of two metallic Ga$_{1-x}$Mn$_{x}$As samples with  $x=4$ \\%, both showing a reasonable agreement between calculated and measured values for the frequency exponent $\\alpha(T)$. The function $g(T)$ and the attempt time $\\tau_0$ are indicated in each case. (c) and (d) Calculated distribution of activation energies $D(E)$ for both samples, showing a similar trend. Local maxima are indicated by black arrows.}\n\\label{PTB_DDH}%\n\\end{figure\n\nSince no signatures of electronic phase separation can be identified for conventional metallic Ga$_{1-x}$Mn$_{x}$As samples with high Mn content ($x=4$ \\%), we next focus on samples irradiated by He ions, whereby disorder in the films is enhanced by the introduction of deep traps, i.e.\\ the Fermi level is shifted by means of carrier compensation in order to change the conductivity from metallic to insulating.     \nFig.$\\,$\\ref{GAMNAS_RES}(a) shows resistivity data of three Ga$_{1-x}$Mn$_{x}$As samples with $x=6$ \\%. Point defects were introduced by the irradiation with an energetic He-ion beam using different doses as described in Section \\ref{growth} (fluences: $2.5\\times 10^{13}\\,$cm$^{-2}$ and $3.5\\times 10^{13}\\,$cm$^{-2}$, hereinafter referred to as \"low dose\" and \"high dose\", respectively). The corresponding Curie temperatures determined from magnetization measurements are marked by arrows. While the unirradiated sample shows the lowest resistivity and metallic behavior with a relatively high $T_\\mathrm{C}$ of $125\\,$K, the resistivity increases strongly as a function of He-ion irradiation dose. At the same time, $T_\\mathrm{C}$ decreases down to $75\\,$K for the sample irradiated with a low He-ion dose and even further to $50\\,$K for a high irradiation dose. The typical maximum in resistivity \\cite{Edmonds2002} becomes less pronounced for more insulating samples. These major changes in resistivity can be explained by an increase of the displacement per atom (DPA) for higher irradiation doses, which results in a decrease of the hole concentration. This is confirmed by Hall effect measurements at room temperature for the three samples (cf.\\ Table \\ref{tab:table1}).\nThe normalized temperature-dependent resistance noise PSD $S_{R}\/R^2$ at $1\\,$Hz for the $x=6$ \\% samples is shown in Fig.$\\,$\\ref{GAMNAS_RES}(b) in a semi-logarithmic plot. The unirradiated sample has the lowest noise level over the entire temperature range. At temperatures below $100\\,$K, the PSD is nearly independent of temperature, followed by a slight increase above $100\\,$K towards room temperature. Likewise, below $80\\,$K, the film irradiated with a low He-ion dose shows the same constant noise magnitude, while there is a much stronger increase of nearly two orders of magnitude towards higher temperatures. The sample irradiated with the high dose shows the highest noise level of all three films below $100\\,$K, also followed by a characteristic increase for $T$ approaching room temperature. The great variation in the PSD for the different samples can be explained by the introduction of deep traps into the As sublattice. An exchange of charge carriers between such traps and the rest of the conducting material, resulting in fluctuations of the hole concentration, can cause the observed $1\/f$-type noise. As shown in previous studies \\cite{Zhou2016}, the concentration of Mn interstitials should not change as a function of irradiation dose, which implies that these interstitials are not the cause for the variation in the PSD as a function of He-ion irradiation dose, although they might still contribute to the $1\/f$ noise. In addition, the noise data shown in Fig.$\\,$\\ref{GAMNAS_RES}(b) suggest a crossover between two temperature regimes: A temperature-independent region below about $100\\,$K and a characteristic increase of the noise magnitude, where the number of activated defects increases towards higher temperatures. The Hooge parameter $\\gamma_{H}$ at room temperature for these samples is of order $\\gamma_{H}=10^{3}$--$10^{5}$ and therefore several orders of magnitude larger than for 'clean' semiconductors. Furthermore, $\\gamma_{H}$ is also several orders of magnitude higher than for the $x=4$ \\% samples, presumably due to the higher Mn content, a higher concentration of traps in the case of the irradiated samples and different growth parameters (substrate temperature and annealing procedure). \\newline \nApart from that, no features around $T_\\mathrm{C}$ were observed except a slight increase below $T_\\mathrm{C}$ for the highest He-ion dose, which might be a hint for weak electronic phase separation or the increasing localization of charge carriers, and a pronounced peak at $80\\,$K. As shown in Ref.\\ \\cite{Lonsky2017}, the application of the phenomenological model by Dutta, Dimon and Horn \\cite{Dutta1979} allows to assign the enhanced noise magnitude at temperatures between 60 and 100\\,K to a distinct peak in the distribution of activation energies $D(E)$ at energies of about $180\\,$meV. This energy can very likely be attributed to traps introduced by the strong He-ion irradiation. No changes in the normalized noise power were found in external magnetic fields up to $7\\,$T (cf.\\ Ref.\\ \\cite{Lonsky2017}). Except the above-mentioned weak increase of the PSD below $T_\\mathrm{C}$ for the sample irradiated with the high dose, there are no indications for an electronic phase separation. As suggested by the large values of $\\gamma_{H}$, possible contributions attributed to a percolative magnetic phase transition may be overshadowed by disorder effects or impurity switching processes. A different approach in order to find indications for a possible electronic phase separation is to tune the Mn content of Ga$_{1-x}$Mn$_{x}$As, which will be discussed in the following section.\n\\begin{figure\n\\centering \t\n\\includegraphics[width=8.5 cm]{HZDR_Heirrad_PSD_AND_Res.eps}\n\\caption{(a) Temperature-dependent resistivity data of three Ga$_{1-x}$Mn$_{x}$As ($x=6$ \\%) samples with different He-ion irradiation doses. (b) Normalized noise PSD as a function of temperature for the three films in a logarithmic representation. No magnetic field dependence or significant features around $T_\\mathrm{C}$ were observed. Arrows indicate the corresponding Curie temperatures.} \n\\label{GAMNAS_RES}%\n\\end{figure}%\n\n\\subsection{Ga\\textsubscript{1-x}Mn\\textsubscript{x}As films with low Mn content and Ga\\textsubscript{1-x}Mn\\textsubscript{x}P}\n\nIn this section, we focus on weakly doped Ga$_{1-x}$Mn$_{x}$As samples with localized charge carriers and compare the results with the metallic films and another insulating Ga$_{1-x}$Mn$_{x}$P reference sample. Fig.\\ \\ref{HZDR_PSD_IN}(a) depicts the resistivity of two as-grown Ga$_{1-x}$Mn$_{x}$As samples with $x=1.8$ \\% and $x=1.2$ \\% and a Ga$_{1-x}$Mn$_{x}$P film with $x=3.5$ \\%. In contrast to the metallic samples, no or only weak features around $T_\\mathrm{C}$ are visible in the resistivity. However, for all three films, the $1\/f$ noise magnitude is significantly enhanced just below the respective Curie temperature, as can be seen in Fig.$\\,$\\ref{HZDR_PSD_IN}(b), where the normalized resistance noise at $1\\,$Hz is plotted versus temperature. For the $x=1.8$ \\% film, only a weak increase occurs around $T_\\mathrm{C}$, followed by a sharp decrease towards lower temperatures. Although this compound is still to be considered as metallic, in the phase diagram it is located very close to the metal-insulator transition (MIT) \\cite{Yuan2017a}, which is why weak signatures of an electronic phase separation are conceivable. The Ga$_{1-x}$Mn$_{x}$As film with $x=1.2$ \\% exhibits nearly the identical noise magnitude between $50$ and $170\\,$K, which is easily comprehensible since the same fabrication technique has been employed, the Mn content is very similar and hence the defect landscape contributing to the resistance noise is comparable. The calculated Hooge parameters at room temperature for both samples are also comparable, namely  $\\gamma_{H}=2\\times 10^{0}$ and $\\gamma_{H}=5\\times 10^{1}$ for the $x=1.2$ \\% and the $x=1.8$ \\% films, respectively. However, the peak in the vicinity of $T_\\mathrm{C}$ is much more pronounced for the $x=1.2$ \\% sample, because this compound is situated right on the edge of the metal-insulator transition, cf.\\ studies of electrical and magnetic properties on the present samples in Ref.\\ \\cite{Yuan2017a}. In detail, this sample still exhibits a global ferromagnetic behavior below $T_\\mathrm{C}$, but electronic transport measurements indicate its insulating character. Within the framework of this study, no noise measurements on Ga$_{1-x}$Mn$_{x}$As samples situated on the insulating side of the MIT could be performed, because the maximum possible current $I$ was not sufficient to measure $1\/f$-type spectra reliably. Instead, the investigated Ga$_{1-x}$Mn$_{x}$P sample is suggested to provide a reference example for the signatures of an electronic phase separation in fluctuation spectroscopy measurements of a diluted magnetic semiconductor with localized charge carriers. For this sample, the Hooge parameter at room temperature amounts to $\\gamma_{H}=1\\times 10^{4}$. As can be seen in Fig.\\ \\ref{HZDR_PSD_IN}(b), the Ga$_{1-x}$Mn$_{x}$P film exhibits a pronounced peak just below $T_\\mathrm{C}$, where the noise level increases by more than one order of magnitude in a small temperature interval.        \nIn analogy to previous studies on the semimetallic ferromagnet EuB\\textsubscript{6} \\cite{Das2012}, the diverging behavior of the resistance noise PSD for the Ga$_{1-x}$Mn$_{x}$P sample can be described by a Lorentz function with a peak at $35.5\\,$K and a width $\\Delta T=2.5\\,$K. In the case of EuB\\textsubscript{6}, Das et al.\\ attribute this sharp peak to a magnetic polaron percolation. As suggested by Kaminski and Das Sarma, such a behavior is also to be expected for Ga$_{1-x}$Mn$_{x}$As or Ga$_{1-x}$Mn$_{x}$P samples with strongly localized charge carriers \\cite{Kaminski2002, Kaminski2003}. Due to the high defect concentration in DMS, it is assumed that the charge carrier concentration is highly inhomogeneous and as ferromagnetism is mediated by charge carriers, upon decreasing the temperature, the ferromagnetic transition will first occur locally within the regions with higher carrier concentration. Upon lowering the temperature, these finite-size clusters will grow and merge until the entire sample becomes ferromagnetic via a percolation transition. \n\\begin{figure\n\\centering\n\\includegraphics[width=8.5 cm]{HZDR_insulsamples_PSD.eps}\n\\caption{(a) Resistivity curves of two Ga$_{1-x}$Mn$_{x}$As samples with low Mn content ($x=1.8$ \\% and $x=1.2$ \\%) and a Ga$_{1-x}$Mn$_{x}$P reference sample ($x=3.5$ \\%). (b) Temperature-dependent noise magnitude of the three samples, all showing an enhanced $1\/f$ noise just below $T_\\mathrm{C}$. Curie temperatures are marked by arrows.} \n\\label{HZDR_PSD_IN}%\n\\end{figure}%\nHowever, in contrast to semimetallic EuB\\textsubscript{6}, where the ferromagnetic transition is accompanied by a drastic reduction of $\\rho(T)$ and a colossal magnetoresistance effect, the temperature dependence of the resistivity of the investigated insulating DMS samples is monotonic. For samples located in the vicinity of the metal-insulator transition, only a small kink is observable in $\\rho(T)$ around $T_\\mathrm{C}$. In general, a ferromagnetic percolation transition is accompanied by an increase of the electrical conductivity, but, at the same time, this may be compensated by the increase of the resistivity with decreasing temperature due to the semiconducting nature of the material \\cite{Kaminski2003}. In the case of strongly localized charge carriers, the decrease in their hopping rate upon decreasing temperature overcomes the decrease in the hopping activation energy due to the ferromagnetic transition. Strikingly, although no features can be observed in the resistivity at $T_\\mathrm{C}$, resistance noise, which is very sensitive to the microscopic current distribution in the sample, shows a strong peak for the Ga$_{1-x}$Mn$_{x}$P sample. Due to the less localized nature of holes in the present weakly doped Ga$_{1-x}$Mn$_{x}$As samples, the possible percolation transition and thus the enhancement of the PSD are less pronounced. In addition, in the vicinity of the maximum in $S_{R}\/R^2$, we find a strong deviation between the calculated frequency exponents from Eq.\\ (\\ref{eq:DDH}) and the experimentally determined values, which is shown in Fig.\\ \\ref{HZDR_GAMNAP_ANALYS}(a) and (b) for the Ga$_{1-x}$Mn$_{x}$As ($x=1.2$ \\%) and the Ga$_{1-x}$Mn$_{x}$P samples, respectively. This non-applicability of the DDH model (marked by gray shaded areas) is another indication for a percolative transition \\cite{Kogan1996}, since the assumptions of this phenomenological approach are not compatible with the nonlinear electronic transport behavior around the percolation threshold. The deviations between calculated and experimentally determined frequency exponents and the divergence in the PSD in the vicinity of the percolation threshold $p_c$ can be understood within the frame of a random resistor network (RRN) model \\cite{Kogan1996}. The reduced number of effective current paths results in the suppression of cancellation of uncorrelated resistance fluctuations along different paths, which are abundant far away from $p_c$ \\cite{Das2012}. Around $p_c$ the current density is strongly inhomogeneous and the most significant contribution to the resistance noise comes from so-called bottlenecks which connect large parts of the infinite cluster. Here, the current density is higher than in other parts of the network.             \nRammal et al.\\ have shown that near the percolation threshold $p_c$, the PSD diverges as $S_{R}\/R^2\\propto(p-p_c)^{-\\kappa}$, while the resistance $R$ behaves as $R\\propto (p-p_c)^{-t}$ \\cite{Rammal1985}. Here, $\\kappa$ and $t$ are critical percolation exponents derived from a RRN model, and $p$ is the fraction of unbroken bonds of a RRN. Due to the non-accessibility of these exponents in an experiment, it is common to link the PSD and the resistance via $S_{R}\/R^2 \\propto R^{w}$ with $w = \\kappa \/t$ \\cite{Yagil1992}. The corresponding analysis for the two relevant thin films is shown in Fig.$\\,$\\ref{HZDR_GAMNAP_ANALYS}(c) and (d), yielding a critical exponent $w=3.7\\pm 0.3$ for Ga$_{1-x}$Mn$_{x}$P and $w=7.1\\pm 0.3$ for Ga$_{1-x}$Mn$_{x}$As ($x=1.2$ \\%). While for Ga$_{1-x}$Mn$_{x}$P this is in fair agreement with typical values for the exponent $w$, e.g.\\ $w=2.9\\pm 0.5$ for perovskite manganites \\cite{Podzorov2000}, the value for the Ga$_{1-x}$Mn$_{x}$As ($x=1.2$ \\%) sample is exceptionally high. We note that no clear systematic changes of shape, position and height of the peak in the temperature-dependent PSD as a function of the applied out-of-plane magnetic field $B$ can be observed. It is assumed that possible changes as a function of the external field are too weak in order to be resolved.             \n\\begin{figure\n\\centering\n\\includegraphics[width=8.5 cm]{HZDR_DDHanalysis_insulating.eps}\n\\caption{(a) and (b) Comparison between experimentally determined and calculated (DDH model) frequency exponents for the Ga$_{1-x}$Mn$_{x}$As sample with $x=1.2$ \\% and the Ga$_{1-x}$Mn$_{x}$P film. Strong deviations occur in the gray shaded areas around the percolative transition. (c) and (d) Log-log plot of PSD versus resistance for Ga$_{1-x}$Mn$_{x}$As ($x=1.2$ \\%)  between $24$ and $32\\,$K and for Ga$_{1-x}$Mn$_{x}$P between $35$ and $45\\,$K. The solid black lines correspond to linear fits yielding $S_{R}\/R^2\\propto R^{w}$ with $w=7.1\\pm 0.3$ for the weakly doped Ga$_{1-x}$Mn$_{x}$As film and  $w=3.7\\pm 0.3$ for the Ga$_{1-x}$Mn$_{x}$P sample.} \n\\label{HZDR_GAMNAP_ANALYS}%\n\\end{figure}%\nWe emphasize that the discussed picture of percolating bound magnetic polarons is expected to be only valid within the impurity-band model and is not compatible with the p-d Zener model, which assumes ferromagnetism being mediated by a Fermi sea of itinerant holes. The impurity-band model suggests that even for strong Mn doping the Fermi energy is located within the separate impurity band and only the degree of localization of the charge carriers will change. \nOur results support the view of holes being trapped in localized impurity band states for weakly doped Ga$_{1-x}$Mn$_{x}$As as well as for Ga$_{1-x}$Mn$_{x}$P, whereas for metallic Ga$_{1-x}$Mn$_{x}$As samples with higher $x$, where no signatures of a percolation mechanism were observed in the electronic noise, the widely-held view of delocalized holes within the valence band mediating ferromagnetism is more appropriate. We note that, for instance, DC transport and optical studies \\cite{Jungwirth2007} as well as first-principle calculations \\cite{Bae2016} corroborate the applicability of the two different models on Ga$_{1-x}$Mn$_{x}$As for the respective Mn concentration ranges. Moreover, because of a strong variation of the Mn energy level among different III-Mn-V combinations, it is unlikely that all materials can be treated within a single model \\cite{Khalid2014}. As a consequence of the higher degree of hole localization in Ga$_{1-x}$Mn$_{x}$P, the peak in $S_{R}\/R^2$ is more pronounced as compared to the insulating Ga$_{1-x}$Mn$_{x}$As sample.            \nAn alternative interpretation of our findings for the Ga$_{1-x}$Mn$_{x}$As with small $x$ valid within the framework of the p-d Zener model could be an electronic phase separation in the vicinity of the metal-insulator transition  \\cite{Ohno2008, Dietl2006}, cf.\\ Section \\ref{intro}. In this case, the nano-scale phase separation results in the coexistence of ferromagnetic bubbles (metallic, hole-rich regime) and a paramagnetic matrix (insulating, hole-poor regime). However, this kind of electronic phase separation should persist in a broad temperature range below $T_\\mathrm{C}$ for insulating Ga$_{1-x}$Mn$_{x}$As samples with low Mn doping, which is not expected to result in such a pronounced and sharp peak in the temperature-dependent noise power spectral density as it was observed in this work. It should also be noted that all investigated samples show global ferromagnetism below $T_\\mathrm{C}$, i.e.\\ there are no mixed phases consisting of ferromagnetic clusters and superparamagnetic grains. Yuan et al.\\ have shown that these mixed phases exist for Ga$_{1-x}$Mn$_{x}$As samples with $x\\leq 0.9$ \\%, but not for $x\\geq 1.2$ \\% \\cite{Yuan2017a}. It is desirable to study the resistance noise behavior of such mixed phases in future.      \n\n\\section{Summary and Conclusion}\nIn this work, we investigated the resistance noise on a series of Ga$_{1-x}$Mn$_{x}$As films with different manganese and defect concentrations and an insulating Ga$_{1-x}$Mn$_{x}$P reference sample, all of which exhibit global ferromagnetism below $T_\\mathrm{C}$. By applying the phenomenological model by Dutta, Dimon and Horn, we calculated the distribution of activation energies $D(E)$ for several samples and discussed a superposition of different types of defects contributing to the measured $1\/f$-type noise. From the comparison of metallic and insulating samples we conclude that resistance noise in metallic Ga$_{1-x}$Mn$_{x}$As samples ($x>2$ \\%) is mainly dominated by impurity switching processes and no prominent features occur around $T_\\mathrm{C}$ even in the presence of an external out-of-plane magnetic field, while insulating samples, in particular Ga$_{1-x}$Mn$_{x}$P, show a sharp peak in the noise magnitude around $T_\\mathrm{C}$ which can be attributed to percolation processes in the material, for which we find a scaling behavior $S_{R}\/R^2 \\propto R^{w}$. Consequently, for Ga$_{1-x}$Mn$_{x}$P, we infer that the picture of percolating magnetic polarons within the impurity-band model is applicable, while for Ga$_{1-x}$Mn$_{x}$As, this picture seems to be valid only for low Mn doping $x$. These findings for samples with localized charge carriers are supported by clear deviations between the calculated (within the DDH model) and experimentally determined frequency exponents $\\alpha$ around the percolative transition. Besides varying the Mn content, another approach to tune Ga$_{1-x}$Mn$_{x}$As samples from the metallic side of the phase diagram towards the insulating regime is to irradiate the films with He ions. It was shown that fluctuation spectroscopy is sensitive to the changes in the defect landscape of irradiated samples, but no clear signs of electronic phase separation could be observed. \\newline\n We have shown that a deeper understanding of defect physics and electronic phase separation in DMS can be obtained from fluctuation spectroscopy measurements. We suggest similar studies on other magnetic semiconductors which are supposed to exhibit percolation transitions or an electronic phase separation, like In$_{1-x}$Mn$_{x}$As or Mn$_{x}$Ge$_{1-x}$ \\cite{Park2002}.\n\n\\section*{Acknowledgements}\nThe ion implantation was done at the Ion Beam Center (IBC) at HZDR. S.\\ Z.\\ acknowledges the financial support by the Helmholtz Association (VH-NG-713). We thank R.\\ P.\\ Campion from the University of Nottingham for supplying the Ga$_{1-x}$Mn$_{x}$As material for the He-ion irradiated samples.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{\\@startsection{section}{0}{\\z@}{5.5ex plus .5ex minus\n 1.5ex}{2.3ex plus .2ex}{\\large\\bf}}\n\\def\\subsection{\\@startsection{subsection}{1}{\\z@}{3.5ex plus .5ex minus\n 1.5ex}{1.3ex plus .2ex}{\\normalsize\\bf}}\n\\def\\subsubsection{\\@startsection{subsubsection}{2}{\\z@}{-3.5ex plus\n-1ex minus  -.2ex}{2.3ex plus .2ex}{\\normalsize\\sl}}\n\n\\renewcommand{\\@makecaption}[2]{%\n   \\vskip 10pt\n   \\setbox\\@tempboxa\\hbox{\\small #1: #2}\n   \\ifdim \\wd\\@tempboxa >\\hsize    \n       \\small #1: #2\\par         \n     \\else                       \n       \\hbox to\\hsize{\\hfil\\box\\@tempboxa\\hfil}\n   \\fi}\n\n\\makeatother\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}