Well-posedness and large deviations for the stochastic Landau Lifshitz Bloch equation

TL;DR

This paper establishes well-posedness and large deviation principles for the stochastic Landau-Lifshitz-Bloch equation with jump noise in dimensions 1, 2, and 3, using advanced probabilistic and analytical methods.

Contribution

It provides the first rigorous proof of existence, uniqueness, and large deviations for solutions of the stochastic Landau-Lifshitz-Bloch equation with jump noise.

Findings

Existence of martingale solutions in all dimensions

Pathwise uniqueness and strong solutions in dimensions 1 and 2

Large deviation principle for small noise asymptotics in dimensions 1 and 2

Abstract

The stochastic Landau-Lifshitz-Bloch equation in dimensions 1; 2; and 3 perturbed by pure jump noise is considered in the Marcus canonical form. A proof for existence of a martingale solution is given. The proof uses the Faedo-Galerkin approximation; which is followed by compactness and tightness arguments. This is followed by use of the Jakubowski's version of the Skorohod Theorem. Pathwise uniqueness and the theory of Yamada and Watanabe give the existence of a strong solution for dimensions 1 and 2. A weak convergence method is later used to establish a Wentzell-Freidlin type large deviation principle for the small noise asymptotic of solutions for dimensions 1 and 2.