Abstract

Abstract: 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 $D$. We show that the map that to the noise associates the unique solution to the equation is locally Lipschitz continuous in the strong norm $L^{\infty}([0,T];H^1(D))\cap L^2([0,T];H^2(D))$, with initial condition in $H^1(D)$. This implies a Wong-Zakai convergence result, a large deviation principle, a support theorem and the Feller property for the associated semigroup.

The talk is based on a joint work with A. Hocquet https://arxiv.org/abs/2103.00926 and on https://arxiv.org/abs/2208.02136.