On the well-posedness of the stochastic Allen-Cahn equation in two dimensions

TL;DR

This paper investigates the mathematical well-posedness of the two-dimensional stochastic Allen-Cahn equation driven by white noise, revealing potential divergence issues in numerical approximations and challenging prior assumptions of well-posedness.

Contribution

It provides a detailed analysis of the ill-posedness of the 2D stochastic Allen-Cahn equation and questions the validity of previous numerical studies claiming well-posedness.

Findings

Numerical approximations diverge as mesh size shrinks.

Regularized noise leads to divergent behavior in the continuum limit.

Published numerical studies may not accurately capture the true behavior of the equation.

Abstract

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical and biological systems in space dimensions d = 1,2,3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d \geq 2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen-Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that a series of published numerical studies are problematic: shrinking the mesh size in these simulations does not lead to the recovery of a physically meaningful limit.