Finite Width For a Random Stationary Interface

Abstract

We study the asymptotic shape of the solution $u(t,x) \in [0,1]$ to a one-dimensional heat equation with a multiplicative white noise term. At time zero the solution is an interface, that is $u(0,x)$ is 0 for all large positive $x$ and $u(0,x)$ is 1 for all large negitive $x$. The special form of the noise term preserves this property at all times $t \geq 0$. The main result is that, in contrast to the deterministic heat equation, the width of the interface remains stochastically bounded.