A monotone convergence theorem for strong Feller semigroups

TL;DR

This paper establishes a monotone convergence theorem for strong Feller semigroups, providing conditions under which the limit of an increasing sequence of such semigroups remains strongly Feller, with applications to elliptic operators.

Contribution

It introduces a new theorem ensuring the strong Feller property persists in limits of semigroups, with implications for elliptic operators with unbounded coefficients.

Findings

The limit semigroup is strongly Feller if the resolvent maps 1 to a continuous function.

The theorem applies to elliptic operators on  with mild regularity assumptions.

Counterexamples show the assumptions are nearly optimal.

Abstract

For an increasing sequence $(T_n)$ of one-parameter semigroups of sub Markovian kernel operators over a Polish space, we study the limit semigroup and prove sufficient conditions for it to be strongly Feller. In particular, we show that the strong Feller property carries over from the approximating semigroups to the limit semigroup if the resolvent of the latter maps the constant 1 function to a continuous function. This is instrumental in the study of elliptic operators on $\mathbb{R}^d$ with unbounded coefficients: our abstract result enables us to assign a semigroup to such an operator and to show that the semigroup is strongly Feller under very mild regularity assumptions on the coefficients. We also provide counterexamples to demonstrate that the assumptions in our main result are close to optimal.