Theory of Probability and Mathematical Statistics
On Poisson equations with a potential in the whole space for "ergodic" generators
Alexander Veretennikov
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Abstract: In \cite{PV01, PV03, PV05, Ver11, KV11} Poisson equation \emph{in the whole space} was studied for so called ergodic generators $L$ corresponding to homogeneous Markov diffusions ($X_t, \, t\ge 0$) in $\mathbb R^d$. Solving this equation is one of the main tools for {\it diffusion approximation} in the theory of stochastic averaging and homogenisation. Here a similar equation {\it with a potential} is considered, firstly because it is natural for PDEs, and secondly with a hope that it may be also useful for some extensions related to homogenization and averaging.
Keywords: SDE, large deviations, Poisson equation, potential, exponential bounds
Bibliography: [1] N.S. Bakhvalov, Averaged characteristics of bodies with periodic structure, Soviet Phys. Dokl., 19, 1974, 650–651.
[2] S.N. Ethier, T.G. Kurtz, Markov Processes: Characterization and Convergence, Wiley Series in Probability and Statistics, 2005 (paperback).
[3] M.A. Krasnosel’skii, E.A. Lifshits, A.V. Sobolev, Positive Linear Systems: The method of positive operators, Berlin, Helderman Verlag, 1989.
[4] N. V. Krylov, On Ito stochastic integral equations, Theory Probab. Appl., 14(2), 1969, 330–336.
[5] N.V. Krylov, Controlled diffusion processes. 2nd ed. Springer Science & Business Media, 2008.
[6] N.V. Krylov, M.V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure, Doklady Mathematics, 245 (1), 1979, 18–20.
[7] A. M. Kulik, A. Yu. Veretennikov, Extended Poisson equation for weakly ergodic Markov processes, Theor. Probability and Math. Statist. 85 (2012), 23–39.
[8] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968.
[9] S.V. Papanicolaou, D. Stroock, S.R.S. Varadhan, Martingale approach to some limit theorems, in: Statistical Mechanics, Dynamical Systems and the Duke Turbulence Conference, Ed. by D. Ruelle, Duke Univ. Math. Series, Vol. 3, Durham, N.C., 1977. ftp://171.64.38.20/pub/papers/papanicolaou/pubs old/martingale duke 77.pdf
[10] A. Piunovskiy, Randomized and relaxed strategies in continuous-time Markov decision processes, SIAM J. Control Optim., 53, 2015, 3503–3533.
[11] ´E. Pardoux, A.Yu. Veretennikov, On the Poisson equation and diffusion approximation. I. Ann. Probab. 29(3), 2001, 1061–1085.
[12] ´E. Pardoux, A.Yu. Veretennikov, On Poisson equation and diffusion approximation 2, Annals of Probability, 31(3), 2003, 1166–1192.
[13] ´E. Pardoux, A.Yu. Veretennikov, On Poisson equation and diffusion approximation 3. Ann. Probab. 33(3), 2005, 1111–1133.
[14] R.T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics and Physics (paperback), 1997.
[15] A.Yu. Veretennikov, Bounds for the Mixing Rate in the Theory of Stochastic Equations, Theory Probab. Appl. 1987, 32, 273–281.
[16] A.Yu. Veretennikov, On large deviations in averaging principle for stochastic differential equations with periodic coefficients. 1. In: Probab. Theory and Math. Statistics. Proc. Fifth Vilnius Conf. (1989). Ed. by B.Grigelionis et al. Vilnius, Lithuania: Mokslas and Utrecht, The Netherlands: VSP, Vol. 2, 542–551.
[17] A.Yu. Veretennikov, On large deviations for diffusion processes with measurable coefficients, Russian Mathematical Surveys, 50(5), 1995, 977–987.
[18] A.Yu. Veretennikov, On polynomial mixing and convergence rate for stochastic difference and differential equations, Theory Probab. Appl. 2001, 45(1), 160–163.
[19] A.Yu. Veretennikov, On Sobolev Solutions of Poisson Equations in Rd with a Parameter (To 70th birthday of Professor N.V. Krylov), Journal of Mathematical Sciences, 179(1), 2011, 48–79.
