Extinction for two parabolic stochastic PDE's on the lattice

It is well known that, starting with finite mass, the super-Brownian motion dies out in finite time. The goal of this article is to show that with som e additional work, one can show finite time die-out for two types of system s of stochastic differential equations on the lattice Z(d) For our first system, let 1/2 less than or equal to gamma < 1, and consider non-negative solutions of du(t, x) = Delta u(t, x) dt + u(gamma) (t, x) dB(x) (t), x epsilon Z(d) u(0, x) = u(0)(x) greater than or equal to 0. Here Delta is the discrete Laplacian and (B-x: x epsilon Z(d)) is a system of independent Brownian motions. We assume that u(0) has finite support. Wh en gamma = 1/2, the measure which puts mass u(t, x) at x is a super-random walk and it is well-known that the process becomes extinct in finite time a .s. Finite-time extinction is known to be a.s. false if gamma = 1. For 1/2 less than or equal to gamma < 1, we show finite-time die-out by breaking up the solution into pieces, and showing that each piece dies in finite time. Unlike the superprocess case, these pieces will not in general evolve inde pendently. Our second example involves the mutually catalytic branching system of stoc hastic differential equations on Z(d). which was first studied in Dawson an d Perkins (1998). dU(t)(x) = Delta U-t(x)dt + root U-t(x)V-t(x) dB(1.x)(t), dV(t)(x) = Delta V-t(x)dt + root U-t(x)V-t(x) dB(2.x)(t), U-0(x) greater than or equal to 0, V-0(x) greater than or equal to 0. By using a somewhat different argument, we show that, depending on the init ial conditions, finite time extinction of one type may occur with probabili ty 0, or with probability arbitrarily close to 1. (C) 2000 Editions scienti fiques et medicales Elsevier SAS.