Triviality of the 2D stochastic Allen-Cahn equation

TL;DR

This paper studies the behavior of solutions to the 2D stochastic Allen-Cahn equation with mollified noise, showing convergence to zero or deterministic solutions depending on noise intensity and mollification removal.

Contribution

It demonstrates the triviality of solutions as mollification is removed and characterizes the limiting behavior under different noise intensity regimes.

Findings

Solutions converge weakly to zero as mollifier is removed.

If noise intensity decreases fast enough, solutions approach the deterministic Allen-Cahn equation.

At a critical noise decay rate, solutions include an additional damping term.

Abstract

We consider the stochastic Allen-Cahn equation driven by mollified space-time white noise. We show that, as the mollifier is removed, the solutions converge weakly to 0, independently of the initial condition. If the intensity of the noise simultaneously converges to 0 at a sufficiently fast rate, then the solutions converge to those of the deterministic equation. At the critical rate, the limiting solution is still deterministic, but it exhibits an additional damping term.