The Annals of Probability

Wright–Fisher diffusion with negative mutation rates

Soumik Pal

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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.

Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 503-526.

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1362750934

Digital Object Identifier
doi:10.1214/11-AOP704

Mathematical Reviews number (MathSciNet)
MR3077518

Zentralblatt MATH identifier
1268.60077

Subjects
Primary: 60G99: None of the above, but in this section 60J05: Discrete-time Markov processes on general state spaces 60J60: Diffusion processes [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

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

Citation

Pal, Soumik. Wright–Fisher diffusion with negative mutation rates. Ann. Probab. 41 (2013), no. 2, 503--526. doi:10.1214/11-AOP704. https://projecteuclid.org/euclid.aop/1362750934


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