Electron. J. Diff. Eqns., Vol. 2000(2000), No. 47, pp. 1-19.

Stochastic perturbations of the Allen-Cahn equation

Tony Shardlow

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.

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Tony Shardlow
Dept. Computer Science
University of Manchester
Oxford Road
Manchester M13 9PL, England
e-amil: shardlow@bioinf.man.ac.uk
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