Invariant measures of L\'evy-type operators and their associated Markov processes

TL;DR

This paper characterizes invariant measures for Markov processes driven by Le9vy-type operators, focusing on integral equations in one dimension and solutions to Le9vy-driven SDEs, with illustrative examples.

Contribution

It introduces a distributional equation criterion for invariant measures of Le9vy-type Markov processes, including a detailed analysis in one dimension and for Le9vy-driven SDEs.

Findings

Distributional equation characterizes invariant measures.

Integral equation form in one-dimensional case.

Examples illustrating the theoretical results.

Abstract

A distributional equation as a criterion for invariant measures of Markov processes associated to L\'evy-type operators is established. This is obtained via a characterization of infinitesimally invariant measures of the associated generators. Particular focus is put on the one-dimensional case where the distributional equation becomes a Volterra-Fredholm integral equation, and on solutions to L\'evy-driven stochastic differential equations. The results are accompanied by various illustrative examples.