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Markov jump processes with a singularity

Part of: Applications - Dynamical systems and ergodic theory Applications to specific physical systems Special processes Markov processes

Published online by Cambridge University Press:  19 February 2016

Ole E. Barndorff-Nielsen ,
Fred Espen Benth  and
Jens Ledet Jensen
Ole E. Barndorff-Nielsen*
Affiliation:
University of Aarhus
Fred Espen Benth*
Affiliation:
University of Aarhus
Jens Ledet Jensen*
Affiliation:
University of Aarhus
*
∗ Postal address: Department of Theoretical Statistics, Departments of Mathematical Sciences and MaPhySto, Ny Munkegade, DK-8000 Aarhus C, Denmark.
∗ Postal address: Department of Theoretical Statistics, Departments of Mathematical Sciences and MaPhySto, Ny Munkegade, DK-8000 Aarhus C, Denmark.
∗ Postal address: Department of Theoretical Statistics, Departments of Mathematical Sciences and MaPhySto, Ny Munkegade, DK-8000 Aarhus C, Denmark.
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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 λ(x) = |x|δ in a neighbourhood of the origin in ℝd 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.

Keywords

Confluent hypergeometric functionlaser coolingrenewal theory

MSC classification

Primary: 60J75: Jump processes
Secondary: 60K05: Renewal theory 81V80: Quantum optics 37N20: Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 779 - 799
Copyright
Copyright © Applied Probability Trust 2000 

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Footnotes

MaPhySto, Centre for Mathematical Physics and Stochastics, funded by a grant from the Danish National Research Foundation.

References

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