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Boundary effect in competition processes

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  01 October 2019

Vadim Shcherbakov  and
Stanislav Volkov
Vadim Shcherbakov*
Affiliation:
Royal Holloway, University of London
Stanislav Volkov*
Affiliation:
Lund University
*
*Postal address: Department of Mathematics, Royal Holloway, University of London, Egham TW20 0EX, UK.
***Postal address: Centre for Mathematical Sciences, Lund University, SE-221 00 Lund, Sweden.
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Abstract

This paper is devoted to studying the long-term behaviour of a continuous-time Markov chain that can be interpreted as a pair of linear birth processes which evolve with a competitive interaction; as a special case, they include the famous Lotka–Volterra interaction. Another example of our process is related to urn models with ball removal. We show that, with probability one, the process eventually escapes to infinity by sticking to the boundary in a rather unusual way.

Keywords

Markov chainbirth-and-death processcompetition processFriedman’s urn modelLyapunov functionmartingale

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 3 , September 2019 , pp. 750 - 768
Copyright
© Applied Probability Trust 2019 

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