Strict continuity of the transition semigroup for the solution of a well-posed martingale problem

TL;DR

This paper establishes a connection between solutions of martingale problems and strongly continuous semigroups on spaces of bounded continuous functions, extending classical results from compact to Polish and locally compact spaces.

Contribution

It extends the classical link between martingale problem solutions and semigroups from compact spaces to Polish and locally compact spaces, including the extension of transition semigroups.

Findings

Connection between martingale solutions and semigroups established

Extension of semigroup theory to Polish spaces

Transition semigroup extension to functions vanishing at infinity

Abstract

In this note we connect the notion of solutions of a martingale problem to the notion of a strongly continuous and locally equi-continuous semigroup on the space of bounded continuous functions equipped with the strict topology. This extends the classical connection of semigroups to Markov processes that was used successfully in the context of compact spaces to the context of Polish spaces. In addition, we consider the context of locally compact spaces and show how the transition semigroup on the space of functions vanishing at infinity can be extended to the space of bounded continuous functions.