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Summary: Digital Object Identifier (DOI) 10.1007/s004409900042
Probab. Theory Relat. Fields 116, 573602 (2000) c Springer-Verlag 2000
C´ecile An´e ˇ Michel Ledoux
On logarithmic Sobolev inequalities
for continuous time random walks on graphs
Received: 6 April 1998 / Revised version: 15 March 1999
Published on line: 14 February 2000
Abstract. We establish modified logarithmic Sobolev inequalities for the path distributions
of some continuous time random walks on graphs, including the simple examples of the
discrete cube and the lattice ZZd
. Our approach is based on the Malliavin calculus on Poisson
spaces developed by J. Picard and stochastic calculus. The inequalities we prove are well
adapted to describe the tail behaviour of various functionals such as the graph distance in
this setting.
1. Introduction
The classical logarithmic Sobolev inequality for Brownian motion B = (Bt )t0 in
IRd [Gr] indicates that for all functionals F in the domain of the Malliavin gradient
operator D : L2( , IP) L2( × [0, T ], IP dt),
IE(F2
log F2
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