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On the large-time behaviour of the solution of a stochastic differential equation driven by a Poisson point process

Part of: Markov processes Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  26 June 2017

Elma Nassar  and
Etienne Pardoux
Elma Nassar*
Affiliation:
Aix-Marseille University
Etienne Pardoux*
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France
*
* Postal address: CMI, Aix-Marseille University, 39 Rue F. Joliot Curie, F-13453 Marseille, France.
** Postal address: CMI, Universite d'Aix Marseille, 39 rue F. Joliot Curie, F-13453 Marseille, France. Email address: [email protected]
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Abstract

We study a stochastic differential equation driven by a Poisson point process, which models the continuous change in a population's environment, as well as the stochastic fixation of beneficial mutations that might compensate for this change. The fixation probability of a given mutation increases as the phenotypic lag Xt between the population and the optimum grows larger, and successful mutations are assumed to fix instantaneously (leading to an adaptive jump). Our main result is that the process is transient (i.e. converges to -∞, so that continued adaptation is impossible) if the rate of environmental change v exceeds a parameter m, which can be interpreted as the rate of adaptation in case every beneficial mutation becomes fixed with probability 1. If v < m, the process is Harris recurrent and possesses a unique invariant probability measure, while in the limiting case m = v, Harris recurrence with an infinite invariant measure or transience depends upon additional technical conditions. We show how our results can be extended to a class of time varying rates of environmental change.

Keywords

Stochastic differential equations with jumpslarge-time behaviourtransiencerecurrence

MSC classification

Primary: 60H10: Stochastic ordinary differential equations 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60G57: Random measures 92D25: Population dynamics (general)
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 344 - 367
Copyright
Copyright © Applied Probability Trust 2017 

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