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Time reversal of some stationary jump diffusion processes from population genetics

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Martin Hutzenthaler  and
Jesse Earl Taylor
Martin Hutzenthaler*
Affiliation:
Goethe-University Frankfurt
Jesse Earl Taylor*
Affiliation:
Arizona State University
*
∗ Current address: LMU Biozentrum, Grosshadern Str. 2, D-82152 Planegg-Martinsried, Germany. Email address: hutzenthaler@bio.lmu.de
∗∗ Postal address: School of Mathematical and Statistical Sciences, Arizona State University, PO Box 871804, Tempe, AZ 85287-1804, USA.
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Abstract

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We describe the processes obtained by time reversal of a class of stationary jump diffusion processes that model the dynamics of genetic variation in populations subject to repeated bottlenecks. Assuming that only one lineage survives each bottleneck, the forward process is a diffusion on [0,1] that jumps to the boundary before diffusing back into the interior. We show that the behavior of the time-reversed process depends on whether the boundaries are accessible to the diffusive motion of the forward process. If a boundary point is inaccessible to the forward diffusion then time reversal leads to a jump diffusion that jumps immediately into the interior whenever it arrives at that point. If, instead, a boundary point is accessible then the jumps off of that point are governed by a weighted local time of the time-reversed process.

Keywords

Time reversaljump diffusionlocal timecoalescentpopulation bottleneckselective sweep

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60J55: Local time and additive functionals 92D10: Genetics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 1147 - 1171
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Research supported by the DFG in the Dutch German Bilateral Research Group Mathematics of Random Spatial Models from Physics and Biology (For 498).

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