Existence of a unique solution and invariant measures for the stochastic Landau–Lifshitz–Bloch equation

Abstract

The Landau–Lifshitz–Bloch equation perturbed by a space-dependent noise was proposed in [10] as a model for evolution of spins in ferromagnetic materials at the full range of temperatures, including the temperatures higher than the Curie temperature. In the case of a ferromagnet filling a bounded domain D⊂Rd, d=1,2,3, we show the existence of strong (in the sense of PDEs) martingale solutions. Furthermore, in cases d=1,2 we prove uniqueness of pathwise solutions and the existence of invariant measures.1