A Poisson equation in $\mathbb{R}^d$ for the elliptic operator
corresponding to an ergodic diffusion process is considered. Existence and
uniqueness of its solution in Sobolev classes of functions is established along
with the bounds for its growth. This result is used to study a diffusion
approximation for two-scaled diffusion processes usingthe method of corrector;
the solution of a Poisson equation serves as a corrector.
References
Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
Bouc, R. and Pardoux, E. (1984). Asymptotic analysis of PDEs with wide-band noise disturbance expansion of the moments. Stochastic Anal. Appl. 2 369-422.
Dynkin, E. B. (1965). Markov Processes 2. Springer, Berlin.
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence, Wiley, New York.
Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer, Berlin.
Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
Khasminski, R.(1966). A limit theorem for solutions of differential equations with random right-hand sides. Theory Probab. Appl. 11 390-406.
Khasminski, R.(1980). Stochastic Stability of Differential Equations. Sijthoff and Nordhoff, The Netherlands.
Krylov, N. V. (1980). Controlled Diffusion Processes (trans. by A. B. Aries). Springer, Berlin.
Ladyzenskaja, O., Solonnikov, V. and Ural'ceva, N. (1968). Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI.
Papanicolaou, G. C., Stroock, D. W. and Varadhan, S. R. S. (1977). Martingale approach to some limit theorems. In Conference on Statistical Mechanics, Dynamical Systems and Turbulence (M. Reed ed.) Duke Univ. Press.
Pardoux, E. and Veretennikov, A. Yu. (1997). Averaging of backward stochastic differential equations, with application to semi-linear PDEs. Stochastics Stochastic. Rep. 60 255- 270.
Revuz, D. (1984). Markov Chains, 2nd rev. ed. North-Holland, Amsterdam.
Stratonovich, R. L. (1963, 1967). Topics in the Theory of Random Noise 1, 2. Gordon and Breach, New York.
Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, Berlin.
Veretennikov, A. Yu. (1982). Parabolic equations and It o's stochastic equations with coefficients discontinuous in the time variable. J. Math. Notes 31 278-283 (trans. from Mat. Zametki 31 549-557).
Veretennikov, A. Yu. (1987). Bounds for the mixingrates in the theory of stochastic equations. Theory Probab. Appl. 32 273-281.
Veretennikov, A. Yu. (1997). On polynomial mixingbounds for stochastic differential equations. Stochastic Process. Appl. 70 115-127.
LATP, UMR-CNRS 6632 Centre de Math´ematiques et d'Informatique Universit´e de Provence 39, rue F. Joliot Curie 13453 Marseille cedex 13 France E-mail: pardoux@cmi.univ-mrs.fr Institute of Information Transmission Problems 19, Bolshoy Karetnii 101447 Moscow Russia E-mail: veretenn@iitp.ru
