Lie-Trotter product formula for locally equicontinuous and tight Markov semigroup

TL;DR

This paper extends the Lie-Trotter product formula to Markov semigroups on measures, demonstrating convergence under local equicontinuity and tightness, thus broadening its applicability beyond classical linear semigroup settings.

Contribution

The paper introduces a Lie-Trotter product formula for Markov semigroups on measure spaces, extending classical results to non-strongly continuous, non-bounded operators.

Findings

Proves convergence of the Lie-Trotter product formula under new conditions.

Establishes a Schur-like property for spaces of measures.

Connects results to classical semigroup theory.

Abstract

In this paper we prove a Lie-Trotter product formula for Markov semigroups in spaces of measures. We relate our results to "classical" results for strongly continuous linear semigroups on Banach spaces or Lipschitz semigroups in metric spaces and show that our approach is an extension of existing results. As Markov semigroups on measures are usually neither strongly continuous nor bounded linear operators for the relevant norms, we prove the convergence of the Lie-Trotter product formula assuming that the semigroups are locally equicontinuous and tight. A crucial tool we use in the proof is a Schur-like property for spaces of measures.