Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 47, pp. 1-19.
Title: Stochastic perturbations of the Allen-Cahn equation
Author: Tony Shardlow (University of Manchester, England)
Abstract:
Consider the Allen-Cahn equation with small diffusion $\epsilon^2$
perturbed by a space time white noise of intensity $\sigma$. In the
limit, $\sigma / \epsilon^2 \rightarrow 0$, solutions converge to
the noise free problem in the $L_2$ norm. Under these conditions,
asymptotic results for the evolution of phase boundaries in the
deterministic setting are extended, to describe the behaviour of the
stochastic Allen-Cahn PDE by a system of stochastic differential
equations. Computations are described, which support the
asymptotic derivation.
Submitted April 18, 2000. Published June 15, 2000.
Math Subject Classifications: 60H15, 74N20, 45M05.
Key Words: dynamics of phase-boundaries; stochastic partial
differential equations; asymptotics.