CPL - Chalmers Publication Library

# The Kac master equation with unbounded collision rate

Bernt Wennberg (Institutionen för matematiska vetenskaper, matematik) ; Mattias Sundén (Institutionen för matematiska vetenskaper, matematisk statistik)
Markov Processes and Related Fields (1024-2953). Vol. 15 (2009), 2, p. 125-148.

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.

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