The Kac Master Equation with Unbounded Collision Rate

#### M. Sunden, B. Wennberg

2009, v.15, Issue 2, 125-148

ABSTRACT

The Kac model is a Markov jump process on the sphere $\sum_{i=1}^N v_i^2 = N$. 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 diffusion 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.

Keywords: Brownian motion,collision kernel,Feller processes,infinitesimal generator,Kac model,Laplace - Beltrami operator,Markov process,semigroup,spectral gap