The Annals of Probability

Wright–Fisher diffusion with negative mutation rates

Soumik Pal

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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.

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

First available in Project Euclid: 8 March 2013

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Zentralblatt MATH identifier

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.)

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


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

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  • [1] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [2] Aldous, D. (1999). Wright–Fisher diffusions with negative mutation rate! Available at
  • [3] Aldous, D. J. (2000). Mixing time for a Markov chain on cladograms. Combin. Probab. Comput. 9 191–204.
  • [4] Barbour, A. D., Ethier, S. N. and Griffiths, R. C. (2000). A transition function expansion for a diffusion model with selection. Ann. Appl. Probab. 10 123–162.
  • [5] Barthe, F. and Wolff, P. (2009). Remarks on non-interacting conservative spin systems: The case of gamma distributions. Stochastic Process. Appl. 119 2711–2723.
  • [6] Durrett, R. (2008). Probability Models for DNA Sequence Evolution, 2nd ed. Springer, New York.
  • [7] Göing-Jaeschke, A. and Yor, M. (2003). A survey and some generalizations of Bessel processes. Bernoulli 9 313–349.
  • [8] Griffiths, R. C. (1979). A transition density expansion for a multi-allele diffusion model. Adv. in Appl. Probab. 11 310–325.
  • [9] Itô, K. and McKean, H. P. Jr. (1974). Diffusion Processes and Their Sample Paths. Springer, Berlin.
  • [10] Lambert, A. (2010). The contour of splitting trees is a Lévy process. Ann. Probab. 38 348–395.
  • [11] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
  • [12] Pal, S. (2011). Analysis of market weights under volatility-stabilized market models. Ann. Appl. Probab. 21 1180–1213.
  • [13] Pal, S. (2011). On the Aldous diffusion on continuum trees. I. Preprint. Available at arXiv:1104.4186v1.
  • [14] Pitman, J. and Yor, M. (1982). A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 59 425–457.
  • [15] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [16] Schweinsberg, J. (2002). An $O(n^{2})$ bound for the relaxation time of a Markov chain on cladograms. Random Structures Algorithms 20 59–70.
  • [17] Yor, M. and Zani, M. (2001). Large deviations for the Bessel clock. Bernoulli 7 351–362.