Stability of invariant measures of the stochastic Landau-Lifshitz-Bloch equation with vanishing noise

TL;DR

This paper studies how invariant measures of the stochastic Landau-Lifshitz-Bloch equation behave as noise vanishes, proving their tightness and convergence to measures of the deterministic system.

Contribution

It establishes the tightness and convergence of invariant measures for the stochastic equation on unbounded domains as noise approaches zero.

Findings

Invariant measures are tight in H^1(R^2) for small noise.

Limit points of invariant measures are invariant measures of the deterministic system.

The approach uses higher-order perturbed viscous systems and tail-end estimates.

Abstract

In this paper, we investigate the limiting dynamics of invariant measures of the stochastic Landau-Lifshitz-Bloch equation driven by the Stratonovich noise defined on the entire space $\R^2$. We first prove the set of all invariant measures of the stochastic equation for small noise is tight in $H^1(\R^2)$, and then prove every limit of a sequence of invariant measures of the stochastic equation must be an invariant measure of the limiting system as the noise intensity approaches zero. The main difficulty of the paper is to establish the tightness of solutions which is caused by the low regularity of solutions and the non-compactness of Sobolev embeddings on unbounded domains. To solve the problem, we first consider a family of higher-order perturbed viscous systems and then use the regularity as well as the uniform tail-ends estimates of the perturbed solutions to establish the tightness of solutions of the original equation by a limiting process.