On the existence and uniqueness of invariant measure for multidimensional diffusion processes

Abstract

This paper deals with the mathematical analysis of multidimensional processes solution of a class of stochastic differential equations. Specifically the analysis is addressed to the derivation of criteria for the existence and uniqueness of the invariant probability measure and its regularity properties in the case of stochastic processes whose infinitesimal generator is uniformly elliptic or degenerate. The criteria are based on the definition of Lyapunov functions and the H\"ormander's rank bracket condition. Finally the criteria are employed for characterizing the invariant probability measure in some applications, including Kolmogorov-Fokker-Planck-type operators.