The stochastic Landau–Lifshitz–Baryakhtar equation: Global solution and invariant measure

Ben Goldys, Agus L. Soenjaya and Thanh Tran

Abstract

The Landau–Lifshitz–Baryakhtar (LLBar) equation perturbed by a space-dependent noise is a system of fourth order stochastic PDEs which models the evolution of magnetic spin fields in ferromagnetic materials at elevated temperatures, taking into account longitudinal damping, long-range interactions, and noise-induced phenomena at high temperatures. In this paper, we show the existence of a martingale solution (which is analytically strong) to the stochastic LLBar equation posed in a bounded domain $\mathcal{D}\subset {\mathbb{R}}^d$, where $d=1,2,3$. We also prove pathwise uniqueness of the solution, which implies the existence of a unique probabilistically strong solution. Finally, we show the Feller property of the Markov semigroup associated with the strong solution, which implies the existence of invariant measures.

Keywords: Fully nonlinear stochastic PDE, Bi-laplacian, tightness of measures, Galerkin approximations, invariant measures.

AMS Subject Classification: Primary 60H15.