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

2011/03/21 by David Applebaum, Applebaum, David · 1 citation
Computer Science · Mathematics · #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #advanced mathematical theories #math.GR #math.PR #msc:60B15

paper · pdf · doi:10.48550/arxiv.1103.4004

arxiv created 2011/03/21 · arxiv updated 2011/03/22

Abstract

We study recurrence and transience for Lévy 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 Lévy 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 Lévy-Khinchine formula.

Cited by

• A Generalised Gangolli-Levy-Khintchine Formula for Infinitely Divisible Measures and Levy Processes on Semi-Simple Lie Groups and Symmetric Spaces

Related

• On Recurrence and Transience of Two-Dimensional Lévy and Lévy-Type Processes
• On Transience of Lévy-Type Processes
• A transience condition for a class of one-dimensional symmetric Lévy processes
• Degrees of transience and recurrence and hierarchical random walks
• Lévy Processes in a Step 3 Nilpotent Lie Group