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Brownian Motion and Levy Processes in Locally Compact David Applebaum
 

Summary: Brownian Motion and L´evy Processes in Locally Compact
Groups
David Applebaum
Probability and Statistics Department,
University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield, England, S3 7RH
e-mail: D.Applebaum@sheffield.ac.uk
Abstract
It is shown that every L´evy process on a locally compact group G is determined by a sequence
of one-dimensional Brownian motions and an independent Poisson random measure. As a conse-
quence, we are able to give a very straightforward proof of sample path continuity for Brownian
motion in G. We also show that every L´evy process on G is of pure jump type, when G is totally
disconnected.
MSC 2000: 60B15,60H20,60G44,22D05
1 Introduction
Let G be a separable locally compact group. A L´evy process on G is essentially a
stochastic process with stationary and independent increments. In the case where G =
Rd
, the study of these is a classical area of investigation for probability theory which

  

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

 

Collections: Mathematics