Extinction for two parabolic stochastic PDE's on the lattice

Abstract

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 some additional work, one can prove finite time die-out for two types of systems of stochastic differential equations on the lattice Z^d. Our first system involves the heat equation on the lattice Z^d, with a nonlinear noise term u(t,x)^gamma dB_x(t), with 1/2 <= gamma < 1. The B_x are independent Brownian motions. When 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 < gamma < 1, we show finite-time die-out by breaking up the solution into pieces, and showing that each piece dies in finite time. Our second example involves the mutually catalytic branching system of stochastic differential equations on Z^d, which was first studied by Dawson and Perkins. Roughly speaking, this process consists of 2 superprocesses with the continuous time simple random walk as the underlying spatial motion. Furthermore, each process stimulates branching and dying in the other process. By using a somewhat different argument, we show that, depending on the initial conditions, finite time extinction of one type may occur with probability 0, or with probability arbitrarily close to 1.