Strong Feller semigroups and Markov processes: A counter example

TL;DR

This paper provides elementary counter-examples demonstrating that strong Feller and joint continuity conditions on Markov semigroups do not guarantee the existence of an associated cdlg Markov process, challenging common assumptions.

Contribution

It introduces simple counter-examples showing the limitations of strong Feller and continuity conditions in ensuring Markov process existence.

Findings

Counter-example with Brownian semigroup on \mathbb{R}\setminus \{0\}\u007F shows no associated cdlg Markov process.

Results extend to right Markov processes even under topology changes with the same Borel sigma-algebra.

Highlights limitations of classical conditions for Markov process construction from semigroups.

Abstract

The aim of this note is to show, by providing an elementary way to construct counter-examples, that the strong Feller and the joint (space-time) continuity for a semigroup of Markov kernels on a Polish space are not enough to ensure the existence of an associated c\`adl\`ag Markov process on the same space. One such simple counter-example is the Brownian semigroup on $\mathbb{R}$ restricted to $\mathbb{R}\setminus \{0\}$, for which it is shown that there is no associated c\`adl\`ag Markov process. Using the same idea and results from potential theory we then prove that the analogous result with c\`adl\`ag Markov process replaced by right Markov process also holds, even if one allows to change the Polish topology to another Polish topology with the same Borel $\sigma$-algebra.