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Markov Processes with Restart

Part of: Groups and semigroups of linear operators, their generalizations and applications Markov processes

Published online by Cambridge University Press:  30 January 2018

Konstantin Avrachenkov ,
Alexey Piunovskiy  and
Yi Zhang
Konstantin Avrachenkov*
Affiliation:
Inria Sophia Antipolis
Alexey Piunovskiy*
Affiliation:
The University of Liverpool
Yi Zhang*
Affiliation:
The University of Liverpool
*
∗ Postal address: Inria Sophia Antipolis, 2004 Route des Lucioles, Sophia Antipolis, 06902, France. Email address: k.avrachenkov@sophia.inria.fr
∗∗ Postal address: Department of Mathematical Sciences, University of Liverpool, M&O Building, L69 7ZL, UK.
∗∗ Postal address: Department of Mathematical Sciences, University of Liverpool, M&O Building, L69 7ZL, UK.
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Abstract

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We consider a general homogeneous continuous-time Markov process with restarts. The process is forced to restart from a given distribution at time moments generated by an independent Poisson process. The motivation to study such processes comes from modeling human and animal mobility patterns, restart processes in communication protocols, and from application of restarting random walks in information retrieval. We provide a connection between the transition probability functions of the original Markov process and the modified process with restarts. We give closed-form expressions for the invariant probability measure of the modified process. When the process evolves on the Euclidean space, there is also a closed-form expression for the moments of the modified process. We show that the modified process is always positive Harris recurrent and exponentially ergodic with the index equal to (or greater than) the rate of restarts. Finally, we illustrate the general results by the standard and geometric Brownian motions.

Keywords

Markov process with restartpositive Harris recurrenceexponential ergodicitystandard and geometric Brownian motions

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 47D07: Markov semigroups and applications to diffusion processes
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 960 - 968
Copyright
© Applied Probability Trust 

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