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

## TL;DR

This paper proves the existence of solutions and invariant measures for the stochastic Landau--Lifshitz--Bloch equation, modeling spin dynamics in ferromagnetic materials at various temperatures, including above the Curie point.

## Contribution

It establishes the existence of strong solutions and invariant measures for the stochastic Landau--Lifshitz--Bloch equation in bounded domains, with uniqueness in lower dimensions.

## Key findings

- Existence of strong martingale solutions in 1, 2, 3 dimensions.
- Uniqueness of solutions in 1 and 2 dimensions.
- Existence of invariant measures in 1 and 2 dimensions.

## Abstract

The Landau--Lifshitz--Bloch equation perturbed by a space-dependent noise was proposed in Garanin 1991 as a model for evolution of spins in ferromagnatic 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 $D\subset \mathbb R^d$, $d=1,2,3$, 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.