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ASPECTS OF RECURRENCE AND TRANSIENCE FOR LÉVY PROCESSES IN TRANSFORMATION GROUPS AND NONCOMPACT RIEMANNIAN SYMMETRIC PAIRS

Part of: Markov processes Topological dynamics Probability theory on algebraic and topological structures Other generalizations Stochastic processes

Published online by Cambridge University Press:  31 May 2013

DAVID APPLEBAUM
DAVID APPLEBAUM*
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, UK email D.Applebaum@sheffield.ac.uk
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Abstract

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We study recurrence and transience for Lévy processes induced by topological transformation groups acting on complete Riemannian manifolds. 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 noncompact type are transient. We show that we also have ‘harmonic transience’, that is, 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.

Keywords

transformation groupLie groupsLévy processrecurrencetransiencepotential measureconvolution semigroupspherical functionGangolli’s Lévy–Khintchine formulatransient Dirichlet spaceharmonic transience

MSC classification

Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60G51: Processes with independent increments; L'evy processes 31C25: Dirichlet spaces 60J45: Probabilistic potential theory 37B20: Notions of recurrence
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 94 , Issue 3 , June 2013 , pp. 304 - 320
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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