Extinction versus exponential growth in a supercritical super-Wright-Fisher diffusion

Klaus Fleischmann and Jan M. Swart

Stochastic Processes and their Applications, 2003, vol. 106, issue 1, 141-165

Abstract: We study mild solutions u to the semilinear Cauchy problem with x[set membership, variant][0,1], f a nonnegative measurable function and [gamma] a positive constant. Solutions to this equation are given by , where is the log-Laplace semigroup of a supercritical superprocess taking values in the finite measures on [0,1], whose underlying motion is the Wright-Fisher diffusion. We establish a dichotomy in the long-time behavior of this superprocess. For [gamma][less-than-or-equals, slant]1, the mass in the interior (0,1) dies out after a finite random time, while for [gamma]>1, the mass in (0,1) grows exponentially as time tends to infinity with positive probability. In the case of exponential growth, the mass in (0,1) grows exponentially with rate [gamma]-1 and is approximately uniformly distributed over (0,1). We apply these results to show that has precisely four fixed points when [gamma][less-than-or-equals, slant]1 and five fixed points when [gamma]>1, and determine their domains of attraction.

Keywords: Binary; splitting; Weighted; superprocess; Semilinear; Cauchy; problem; Finite; ancestry; property; Trimmed; tree; Compensated; h-transform (search for similar items in EconPapers)
Date: 2003
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304-4149(03)00043-7
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:106:y:2003:i:1:p:141-165

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().