Mutually Catalytic Branching in The Plane: Infinite Measure States
Donald A. Dawson, Carleton University
Alison M. Etheridge, University of Oxford
Klaus Fleischmann, Weierstrass Institute for Applied Analysis and Stochastics
Leonid Mytnik, Technion - Israel Institute of Technology
Edwin A. Perkins, The University of British Columbia
Jie Xiong, University of Tennessee
Abstract
A two-type
infinite-measure-valued population in R^2 is constructed which undergoes
diffusion and branching. The system is interactive in that the branching rate
of each type is proportional to the local density of the other type. For a
collision rate sufficiently small compared with the diffusion rate, the model
is constructed as a pair of infinite-measure-valued processes which satisfy a
martingale problem involving the collision local time of the solutions. The
processes are shown to have densities at fixed times which live on disjoint
sets and explode as they approach the interface of the two populations. In the
long-term limit (in law), local extinction of one type is shown. Moreover the
surviving population is uniform with random intensity. The process constructed
is a rescaled limit of the corresponding Z^2-lattice model studied by
Dawson and Perkins (1998) and resolves the large scale mass-time-space behavior
of that model under critical scaling. This part of a trilogy extends results
from the finite-measure-valued case, whereas uniqueness questions are again
deferred to the third part.
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