A Note on Evolution Systems of Measures for Time-Dependent Stochastic Differential Equations

Abstract

We consider a stochastic equation in ℝⁿ with time-dependent coefficients assuming that it has a unique solution and denote by P _s,t , s < t the corresponding transition semigroup. Then we consider a family of measures (ν _t ) _t∈ℝ such that $\smallint _{\mathbb{R}^d } P_{s,t} \varphi \left( x \right)\nu _s \left( {dx} \right) = \smallint _{\mathbb{R}^d } \varphi \left( x \right)\nu _t \left( {dx} \right),s \leqslant t$ , for all continuous and bounded functions ϕ. The family (ν _t ) _t∈ℝ is called an evolution system of measures indexed by ℝ. It plays the role of a probability invariant measure for autonomous systems. In this paper we generalize the Krylov-Bogoliubov criterion to prove the existence of an evolution system of measures. Moreover, we study some properties of the corresponding Kolmogorov operator proving in particular that it is dissipative with respect to the measure ν(dt, dx) = ν _t (dx)dt.

The first author would like to thank the University of Bielefeld for its kind hospitality and financial support. This work was also supported by the research program “Equazioni di Kolmogorov” from the Italian “Ministero della Ricerca Scientifica e Tecnologica”.

The second-named author would like to thank the Scuola Normale Superiore for a very pleasant stay in Pisa during which most of this work was done. Financial support of the SNS as well as of the DFG-Forschergruppe “Spectral Analysis, Asymptotic Distributions, and Stochastic Dynamics” is gratefully acknowledged.