Aspects of Recurrence and Transience for Levy Processes in Transformation Groups and Non-Compact Riemannian Symmetric Pairs

TL;DR

This paper investigates recurrence and transience properties of Levy processes on transformation groups and symmetric spaces, establishing criteria based on potential measures and characteristic exponents, with implications for harmonic analysis.

Contribution

It introduces new criteria for transience and recurrence of Levy processes in symmetric spaces, linking potential measures and harmonic transience to the process characteristics.

Findings

Transience characterized by finite potential measure on compact sets.

All bi-invariant Levy processes in irreducible non-compact symmetric pairs are transient.

Harmonic transience linked to local integrability of the inverse of the real part of the characteristic exponent.

Abstract

We study recurrence and transience for L\'{e}vy processes induced by topological transformation groups. In particular the transience-recurrence dichotomy in terms of potential measures is established and transience is shown to be equivalent to the potential measure having finite mass on compact sets when the group acts transitively. It is known that all bi-invariant L\'{e}vy processes acting in irreducible Riemannian symmetric pairs of non-compact type are transient. We show that we also have "harmonic transience", i.e. local integrability of the inverse of the real part of the characteristic exponent which is associated to the process by means of Gangolli's L\'{e}vy-Khinchine formula.