Motion of a Droplet for the Stochastic Mass Conserving Allen Cahn Equation
We study the stochastic mass conserving Allen Cahn equation posed on a smoothly bounded domain of \mathbb{R}^2$ with additive, spatially smooth, space time noise. This equation describes the stochastic motion of a small almost semicircular droplet attached to domain's boundary and moving toward a point of locally maximum curvature. We apply Ito calculus to derive the stochastic dynamics of the center of the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen, and Fusco [Calc. Var. 11 (2000) pp. 233 305] for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption of a sufficiently small noise strength, we establish stochastic stability of a neighborhood of the manifold of boundary droplet states in the $L^2$ and $H^1$ norms, which means that with overwhelming probability the solution stays close to the manifold for very l.
SIAM J. Math. Anal.