## The Annals of Probability

### On the Poisson Equation and Diffusion Approximation. I

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 29, Number 3 (2001), 1061-1085.

Dates
First available in Project Euclid: 5 March 2002

https://projecteuclid.org/euclid.aop/1015345596

Digital Object Identifier
doi:10.1214/aop/1015345596

Mathematical Reviews number (MathSciNet)
MR1872736

Zentralblatt MATH identifier
1029.60053

#### Citation

Pardoux, E.; Veretennikov, Yu. On the Poisson Equation and Diffusion Approximation. I. Ann. Probab. 29 (2001), no. 3, 1061--1085. doi:10.1214/aop/1015345596. https://projecteuclid.org/euclid.aop/1015345596

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• 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