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The Kac master equation with unbounded collision rate

Journal article
Authors Bernt Wennberg
Mattias Sundén
Published in Markov Processes and Related Fields
Volume 15
Issue 2
Pages 125-148
ISSN 1024-2953
Publication year 2009
Published at Department of Mathematical Sciences, Mathematical Statistics
Department of Mathematical Sciences, Mathematics
Pages 125-148
Language en
Keywords Brownian motion, collision kernel, Feller processes, innitesimal generator, Kac model, Laplace - Beltrami operator, Markov process, semigroup, spectral gap
Subject categories Mathematical Analysis, Mathematical statistics, Other Mathematics


The Kac model is a Markov jump process on the sphere $\sum_{j=1}^{N} v_j^2$. The model was conceived as model for an N-particle system with pairwise interactions, and hence the jumps involve only pairs of coordinates, $(v_i, v_j )$. This paper deals with Kac models with unbounded jump rates. We prove that the processes are Feller processes, and introduce a diusion approximation that is useful for numerical simulation of the processes. We also study the spectral gap of the Markov generators, using the methods developed by Carlen, Carvalho and Loss.

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