On the well-posedness of the stochastic Allen-Cahn equation in two dimensions
This work challenges the validity of existing numerical simulations for a widely used SPDE model in two dimensions, highlighting a critical gap between applied practice and mathematical theory.
The paper investigates the well-posedness of the two-dimensional stochastic Allen-Cahn equation with additive white noise, providing heuristic and numerical evidence that approximations diverge in the continuum limit, suggesting that previous numerical studies may not converge to a physically meaningful limit.
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.