On the Poisson equation and diffusion approximation. I
type: Item
instance of: scholarly article
title: On the Poisson equation and diffusion approximation. I
zbMATH Open document ID: 1029.60053
author: Etienne Pardoux
author: A. Yu. Veretennikov
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: The authors consider the Poisson equation \(Lu=-f\) in the entire \(d\)-dimensional space. \(L\) denotes a second-order elliptic differential operator which is the generator of a positive recurrent diffusion process \(X\). The function \(f\) is assumed to be centred with respect to the invariant measure of \(X\). The first part of the paper investigates under what conditions the function \(u(x)=\int_0^\infty E_x[f(X_t)]dt\) defines a solution of the above Poisson equation. The two main conditions are that the diffusion coefficient is strongly elliptic and that the drift coefficient \(b\) satisfies the mixing condition \(\langle b(x),x/|x|\rangle\leq -r |x|^\alpha\) for certain \(r>0\) and \(\alpha\geq -1\) outside a compact set. Several properties of the solution are derived by probabilistic methods. The second part applies these results to singularly perturbed random differential equations and establishes their convergence to a stochastic differential equation. In the appendix a version of the Itô-Krylov formula is proved.
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 1906011
zbMATH Keywords: Poisson equation
zbMATH Keywords: invariant measure
zbMATH Keywords: polynomial recurrence
zbMATH Keywords: diffusion approximation
zbMATH Keywords: Itô-Krylov formula
reviewed by: Markus Reiß
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1015345596
OpenAlex ID: W4238063566
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DOI: 10.1214/AOP/1015345596
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description: scientific article; zbMATH DE number 1906011