Vol. 84, No. 1, 1979

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Stochastic diffusion on an unbounded domain

Robert Marcus

Vol. 84 (1979), No. 1, 143–153

In this paper we study a stochastic partial equation of the following form.

∂u-     ∂2u-
∂t = 1∕2∂x2 − f(u)+ α(x,t)

where f is a monotone nonlinear operator and α is a “white noise” process in x and t. In a previous paper we demonstrated the existence of a unique solution in a generalized sense for x in a bounded domain. This solution was decomposed into the sum of a stationary process and a transient process. An explicit representation was found for the stationary distribution of the stationary process. If f is an ordinary function of u(x) then the stationary distribution is associated with a Markov process in x. The purpose of this paper is to remove the restriction of boundedness for the bounded domain.

Mathematical Subject Classification 2000
Primary: 60H15
Secondary: 35R60, 60J60, 81E99
Received: 9 May 1978
Published: 1 September 1979
Robert Marcus