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March 2013 Wright–Fisher diffusion with negative mutation rates
Soumik Pal
Ann. Probab. 41(2): 503-526 (March 2013). DOI: 10.1214/11-AOP704

Abstract

We study a family of $n$-dimensional diffusions, taking values in the unit simplex of vectors with nonnegative coordinates that add up to one. These processes satisfy stochastic differential equations which are similar to the ones for the classical Wright–Fisher diffusions, except that the “mutation rates” are now nonpositive. This model, suggested by Aldous, appears in the study of a conjectured diffusion limit for a Markov chain on Cladograms. The striking feature of these models is that the boundary is not reflecting, and we kill the process once it hits the boundary. We derive the explicit exit distribution from the simplex and probabilistic bounds on the exit time. We also prove that these processes can be viewed as a “stochastic time-reversal” of a Wright–Fisher process of increasing dimensions and conditioned at a random time. A key idea in our proofs is a skew-product construction using certain one-dimensional diffusions called Bessel-square processes of negative dimensions, which have been recently introduced by Göing-Jaeschke and Yor.

Citation

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Soumik Pal. "Wright–Fisher diffusion with negative mutation rates." Ann. Probab. 41 (2) 503 - 526, March 2013. https://doi.org/10.1214/11-AOP704

Information

Published: March 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1268.60077
MathSciNet: MR3077518
Digital Object Identifier: 10.1214/11-AOP704

Subjects:
Primary: 60G99 , 60J05 , 60J60 , 60J80

Keywords: Bessel processes of negative dimension , Continuum random tree , Markov chain on cladograms , Wright–Fisher diffusion

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 2 • March 2013
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