Asymptotic stability for a class of Markov semigroups

TL;DR

This paper establishes conditions under which a Markov semigroup on a compact metric space exhibits asymptotic stability, ensuring convergence of the semigroup to a unique invariant function.

Contribution

It introduces new criteria involving barriers and maximum principles that guarantee the asymptotic stability of Markov semigroups on compact spaces.

Findings

Existence of a uniform limit for the semigroup applied to any continuous function.

The limit function is continuous and invariant under the semigroup.

The limit agrees with the initial function on the boundary of the open dense subset.

Abstract

Let $U\subset K$ be an open and dense subset of a compact metric space and let $\{\Phi_t\}_{t\ge0}$ be a Markov semigroup on the space of bounded Borel measurable functions on $U$ with the strong Feller property. Suppose that for each $x\in\bdu$ there exists a barrier $h\in C(K)$ at $x$ such that $\Phi_t(h)\ge h$ for all $t\ge0$. Suppose also that every real-valued $g\in C(K)$ with $\Phi_t(g)\ge g$ for all $t\ge0$ and which attains its global maximum at a point inside $U$ is constant. Then for each $f\in C(K)$ there exists the uniform limit $F=\lim_{t\to\infty}\Phi_t(f)$. Moreover $F$ is continuous on $K$, agrees with $f$ on $\partial{U}$ and $\Phi_t(F)=F$ for all $t\ge0$.