A Journal "Theory of Probability and Mathematical Statistics"
2021
2020
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970

Archive

# Theory of Probability and Mathematical Statistics

## On Poisson equations with a potential in the whole space for "ergodic" generators

### Alexander Veretennikov

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.