Strong convergence of solutions to nonautonomous Kolmogorov equations

TL;DR

This paper investigates the long-term behavior of solutions to nonautonomous Kolmogorov equations with unbounded coefficients, proving convergence to constants and establishing measure uniqueness.

Contribution

It provides new results on the strong convergence of solutions and the uniqueness of evolution systems of measures for these equations.

Findings

Solutions converge to constant functions as time approaches infinity.

Uniqueness of the tight evolution system of measures is established.

Results include the case where coefficients converge over time.

Abstract

We study a class of nonautonomous, linear, parabolic equations with unbounded coefficients on $\mathbb R^{d}$ which admit an evolution system of measures. It is shown that the solutions of these equations converge to constant functions as $t\to+\infty$. We further establish the uniqueness of the tight evolution system of measures and treat the case of converging coefficients.