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Invariant measures for Lévy flows of diffeomorphisms

Published online by Cambridge University Press:  11 July 2007

David Applebaum  and
Hiroshi Kunita
David Applebaum
Affiliation:
Department of Mathematics, Statistics and Operational Research, Nottingham Trent University, Burton Street, Nottingham NG1 4BU, UK
Hiroshi Kunita
Affiliation:
Graduate School of Mathematics, Kyushu University 36, Fukuoka 812, Japan
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Abstract

We consider the Lévy flow of diffeomorphisms of a manifold obtained by solving a stochastic differential equation driven by an n-dimensional Lévy process and n complete smooth vector fields (which generate a finite-dimensional Lie algebra L) and show that a necessary and sufficient condition for a large class of such flows to have an invariant measure is that each of the vector fields is divergence-free. We investigate the existence and uniqueness of such measures and examine their invariance with respect to the action of the associated Markov semigroup. In particular, we prove that there is an invariant probability measure on M if and only if the transformation Lie group G whose Lie algebra is L is compact. In the unbounded case we show that if there is a unique G-invariant measure, then it is also the unique invariant measure for the Markov semigroup provided the flow is recurrent.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 130 , Issue 5 , October 2000 , pp. 925 - 946
Copyright
Copyright © Royal Society of Edinburgh 2000

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