Weak solutions of the stochastic Landau–Lifshitz–Gilbert equation

We consider a Landau-Lifshitz-Gilber equation perturbed by a multiplicative space-dependent noise for a ferromagnet filling a bounded three-dimensional domain. We show the existence of weak martingale solutions taking values in a sphere S2. The reg-ularity of weak solutions is also discussed. Our research is in response to the paper by Kohn et al. “Magnetic elements at finite temperature and large deviation theory. ” Journal of Nonlinear Science 15 (2005): 223–53, which calls for the study of stochastic equations with spatially varying magnetization.