An application of rough paths theory to the study of the
stochastic Landau-Lifschitz-Gilbert equation
The Landau-Lifschitz-Gilbert equation is a model describing the magnetisation of a ferromagnetic material. The stochastic model is studied to observe the role of thermal fluctuations. We interpret the linear multiplicative noise appearing by means of rough paths theory and we study existence and uniqueness of the solution to the equation on a one dimensional domain D. As a consequence of the rough paths formulation, the map that to the noise associates the unique solution to the equation is locally Lipschitz continuous in the strong norm L^\infty([0,T];H^1(D))\cap L^2([0,T];H^2(D)). This implies a Wong-Zakai convergence result, a large deviation principle, a support theorem. We prove also continuity with respect to the initial datum in H^1(D), which allows to conclude the Feller property for the associated semigroup. We also discuss briefly a pathwise central limit theorem and a moderate deviations principle for the stochastic LLG: in this case the rough paths formulation leads to a pathwise convergence, not easily reachable in the classical Ito’s calculus setting. The talk is based on a joint work with A. Hocquet [https://doi.org/10.1016/j.jfa.2023.110094], on [https://arxiv.org/abs/2208.02136] and on [https://arxiv.org/abs/2307.10965].
The talk will be followed by refreshments in the Common Room at 4pm.