Feller Evolution Systems: Generators and Approximation

TL;DR

This paper studies Feller evolution systems, focusing on their generators and how they can be approximated by Markov chains with Lévy increments, revealing new insights into their structure and approximation methods.

Contribution

It introduces conditions under which Feller evolutions can be approximated by Markov chains with Lévy increments and explores the generator's properties, including cases with discontinuous symbols.

Findings

Feller evolution corresponds to a higher-dimensional Feller process.

Approximation of Feller evolutions by Markov chains is possible under mild conditions.

Generators may have discontinuous symbols in general.

Abstract

A time and space inhomogeneous Markov process is a Feller evolution process, if the corresponding evolution system on the continuous functions vanishing at infinity is strongly continuous. We discuss generators of such systems and show that under mild conditions on the generators a Feller evolution can be approximated by Markov chains with L\'evy increments. The result is based on the approximation of the time homogeneous space-time process corresponding to a Feller evolution process. In particular, we show that a $d$-dimensional Feller evolution corresponds to a $d+1$-dimensional Feller process. It is remarkable that, in general, this Feller process has a generator with discontinuous symbol.