Pseudo Differential Operators and Markov Semigroups on Compact Lie Groups

TL;DR

This paper develops a framework using pseudo-differential operators on compact Lie groups to analyze Markov processes, linking Fourier analysis with stochastic processes and providing explicit examples involving Lévy processes.

Contribution

It extends the Ruzhansky-Turunen theory to study group-valued Markov processes and characterizes Feller semigroups as pseudo-differential operators on compact Lie groups.

Findings

Feller semigroups are represented as pseudo-differential operators with explicit symbols.

Explicit computation of symbols for Lévy-driven stochastic flows on groups.

Conditions established for Lévy-type operators to generate symmetric Dirichlet forms.

Abstract

We extend the Ruzhansky-Turunen theory of pseudo differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N.Jacob and collaborators. Feller semigroups, their generators and resolvents are exhibited as pseudo-differential operators and the symbols of the operators forming the semigroup are expressed in terms of the Fourier transform of the transition kernel. The symbols are explicitly computed for some examples including the Feller processes associated to stochastic flows arising from solutions of stochastic differential equations on the group driven by L\'{e}vy processes. We study a family of L\'{e}vy-type linear operators on general Lie groups that are pseudo differential operators when the group is compact and find conditions for them to give rise to symmetric Dirichlet forms.