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On the Poisson equation and diffusion approximation. I. (English) Zbl 1029.60053
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.

MSC:
60H30 Applications of stochastic analysis (to PDEs, etc.)
35J15 Second-order elliptic equations
60J45 Probabilistic potential theory
60J60 Diffusion processes
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[19] LATP, UMR-CNRS 6632 Centre de Mathématiques et d’Informatique Université 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
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