Invariant measures for systems of Kolmogorov equations

TL;DR

This paper establishes conditions for the existence of invariant measures in systems of Kolmogorov equations, showing their properties and role in describing long-term behavior of associated semigroups.

Contribution

It provides new sufficient conditions for invariant measures in systems of parabolic PDEs with unbounded coefficients, including their absolute continuity and asymptotic significance.

Findings

Invariant measures exist under specified conditions.

Measures are absolutely continuous with respect to Lebesgue measure.

They characterize the semigroup's asymptotic behavior.

Abstract

In this paper we provide sufficient conditions which guarantee the existence of a system of invariant measures for semigroups associated to systems of parabolic differential equations with unbounded coefficients. We prove that these measures are absolutely continuous with respect to the Lebesgue measure and study some of their main properties. Finally, we show that they characterize the asymptotic behaviour of the semigroup at infinity.