Singular limits for stochastic equations

TL;DR

This paper investigates the singular limits of stochastic evolution equations, focusing on the interplay of diminishing noise and operator vanishing, revealing how deterministic terms emerge through renormalization.

Contribution

It introduces a general framework for analyzing singular limits in stochastic equations, including error estimates and applications to complex models like Allen-Cahn and Cahn-Hilliard equations.

Findings

Recovered known results on vanishing noise with increasing roughness.

Established a new framework for error analysis in singular limits.

Applied results to specific models with space-time white noise.

Abstract

We study singular limits of stochastic evolution equations in the interplay of disappearing strength of the noise and insufficient regularity, where the equation in the limit with noise would not be defined due to lack of regularity. We recover previously known results on vanishing small noise with increasing roughness, but our main focus is to study for fixed noise the singular limit where the leading order differential operator in the equation may vanish. Although the noise is disappearing in the limit, additional deterministic terms appear due to renormalization effects. We separate the analysis of the equation from the convergence of stochastic terms and give a general framework for the main error estimates. This first reduces the result to bounds on a residual and in a second step to various bounds on the stochastic convolution. Moreover, as examples we apply our result to the a singularly regularized Allen-Cahn equation with a vanishing Bilaplacian, and the Cahn-Hilliard/Allen-Cahn homotopy with space-time white noise in two spatial dimensions.