MaPhySto
Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

MPS-RR 1999-42
November 1999

# Markov jump processes with a singularity

by:

## Abstract

Certain types of Markov jump processes x(t) with continuous state space and one or more absorbing states are studied. Cases where the transition rate in state x is of the form \lambda(x) = |x|^\delta in a neighbourhood of the origin in R^d are considered, in particular. This type of problem arises from quantum physics in the study of laser cooling of atoms, and the present paper connects to recent work in the physics literature. The main question addressed is that of the asymptotic behaviour of x(t) near the origin for large t. The study involves solution of a renewal equation problem in continuous state space.

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This paper has now been published in Ann. Appl. Prob. 32 (2000), 779-799.