A pathwise stochastic Landau-Lifshitz-Gilbert equation with application to large deviations

TL;DR

This paper develops a pathwise stochastic formulation of the Landau-Lifshitz-Gilbert equation using rough path theory, establishing existence, uniqueness, regularity, and large deviation principles for solutions with multidimensional noise.

Contribution

It introduces a direct rough path approach to the stochastic Landau-Lifshitz-Gilbert equation, enabling analysis without transformations and handling multidimensional noise.

Findings

Proved existence and uniqueness of solutions in energy spaces.

Established continuity of the Itô-Lyons map in optimal norms.

Derived Wong-Zakai approximation, large deviation principle, and support theorem.

Abstract

Using a rough path formulation, we investigate existence, uniqueness and regularity for the stochastic Landau-Lifshitz-Gilbert equation with Stratonovich noise on the one dimensional torus. As a main result we show the continuity of the so-called It\^o-Lyons map in the energy spaces $L^\infty(0,T;H^k)\cap L^2(0,T;H^{k+1})$ for any $k\ge1$. The proof proceeds in two steps. First, based on an energy estimate in the aforementioned space together with a compactness argument we prove existence of a unique solution, implying the continuous dependence in a weaker norm. This is then strengthened in the second step where the continuity in the optimal norm is established through an application of the rough Gronwall lemma. Our approach is direct and does not rely on any transformation formula, which permits to treat multidimensional noise. As an easy consequence we then deduce a Wong-Zakai type result, a large deviation principle for the solution and a support theorem.