  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.
