Polynomial weak approximation for stochastic reaction-diffusion equations near the sharp interface limit

Abstract

We study weak approximation for stochastic reaction-diffusion equations in the sharp-interface regime, where the diffuse interface thickness \(ε\) is small and the dependence of numerical constants on \(ε^{-1}\) is a central issue. For an additive-noise stochastic Allen--Cahn type equation, direct stability arguments typically produce weak error bounds with constants growing exponentially in \(ε^{-1}\). Such estimates do not capture the polynomial stability expected near the sharp interface limit. We prove polynomial-in-\(ε^{-1}\) weak error bounds for an accelerated splitting exponential Euler approximation. The proof combines time-uniform moment estimates, regularity estimates for the exact and numerical dynamics, and time-independent derivative estimates for the Kolmogorov semigroup. The key point is that the averaged regularizing effect of the noise, expressed through asymptotic strong Feller or strong Feller estimates, replaces the deterministic spectral estimate for the linearized Allen--Cahn operator. The result gives a weak approximation theory whose constants depend polynomially on \(ε^{-1}\) and explicitly on the covariance regularity and non-degeneracy parameters.

33 pages