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Aspects of Recurrence and Transience for Levy Processes in Transformation Groups and

Summary: Aspects of Recurrence and Transience for L´evy
Processes in Transformation Groups and
Non-Compact Riemannian Symmetric Pairs
David Applebaum,
School of Mathematics and Statistics,
University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield, England, S3 7RH
e-mail: D.Applebaum@sheffield.ac.uk
We study recurrence and transience for L´evy 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´evy processes acting in irreducible Riemannian sym-
metric 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´evy-Khinchine formula.


Source: Applebaum, David - Department of Probability and Statistics, University of Sheffield


Collections: Mathematics