# The Kac master equation with unbounded collision rate Artikel i vetenskaplig tidskrift, 2009

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.

spectral gap

innitesimal generator

Kac model

Brownian motion

collision kernel

semigroup

Laplace - Beltrami operator

Markov process

Feller processes

## Författare

#### Bernt Wennberg

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

#### Mattias Sunden

Göteborgs universitet

Chalmers, Matematiska vetenskaper, matematisk statistik

#### Markov Processes and Related Fields

1024-2953 (ISSN)

Vol. 15 2 125-148

#### Ämneskategorier

Annan matematik

Sannolikhetsteori och statistik

Matematisk analys

2017-10-07