WIAS Preprint No. 752, (2002)

Extinction versus explosion in a supercritical super-Wright-Fisher diffusion



Authors

  • Fleischmann, Klaus
  • Swart, Jan

2010 Mathematics Subject Classification

  • 60J80 60G57 35K55 35K15 60J57 60J60

Keywords

  • Binary splitting, binary branching, weighted superprocess, semilinear Cauchy problem, semilinear parabolic PDE, finite ancestry property, trimmed tree, h-transform, extinction, explosion

Abstract

We study mild solutions to a semilinear Cauchy problem related to a supercritical superprocess taking values in the finite measures on the unit interval, whose underlying motion is the Wright-Fisher diffusion. We establish a dichotomy in the long-time behavior of this superprocess. When a parameter related to the growth of the process is smaller than or equal to a certain critical value (in our case one), the mass in the interior of the unit interval dies out after a finite random time, while for larger values of the growth parameter, the mass in the interior explodes with positive probability as time tends to infinity. In the case of explosion, the mass in the interior grows exponentially and is approximately uniformly distributed over the unit interval. We apply these results to show that the semilinear Cauchy problem has precisely four fixed points when the growth parameter is small and five fixed points when the growth parameter is large, and determine their domains of attraction.

Appeared in

  • Stoc. Proc. Appl., vol.106, number 1, pp. 141-165, under new title: Extinction versus Exponential Growth in a Supercritical Super-Wright- Fischer Diffusion

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