Invariant measures and their limiting behaviour of the Landau-Lifshitz-Bloch equation in unbounded domains

Abstract This paper deals with the existence and limiting behaviour of invariant measures of the stochastic Landau–Lifshitz–Bloch equation driven by linear multiplicative noise and additive noise defined in the entire space R d for d = 1 , 2 , which describes the phase spins in ferromagnetic materials around the Curie temperature. We first establish the existence and uniqueness of solutions by a domain expansion method. We then prove the existence of invariant measures by the weakly Feller argument. In the case d = 1, we show the uniform tightness of the set of all invariant measures of the stochastic equation, and prove any limit of a sequence of invariant measures of the perturbed equation must be an invariant measure of the limiting system. The cut-off arguments, stopping time techniques and uniform tail-ends estimates of solutions are developed to overcome the difficulty caused by the high-order nonlinearity and the non-compactness of Sobolev embeddings in unbounded domains.