Geodesic
Stochastic perturbation of the Allen-Cahn equation
Electronic journal of differential equations, Tome 2000 (2000)
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.
Classification : 60H15, 74N20, 45M05
Keywords: dynamics of phase-boundaries, stochastic partial differential equations, asymptotics 
@article{EJDE_2000__2000__a136,
     author = {Shardlow,  Tony},
     title = {Stochastic perturbation of the {Allen-Cahn} equation},
     journal = {Electronic journal of differential equations},
     year = {2000},
     volume = {2000},
     zbl = {0959.60047},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2000__2000__a136/}
}
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%J Electronic journal of differential equations
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Shardlow,  Tony. Stochastic perturbation of the Allen-Cahn equation. Electronic journal of differential equations, Tome 2000 (2000). http://geodesic.mathdoc.fr/item/EJDE_2000__2000__a136/