No abstract

We propose a new convergent time semi-discrete scheme for the stochastic Landau-Lifshitz-Gilbert equation. The scheme is only linearly implicit and does not require the resolution of a nonlinear problem at each time step. Using a martingale approach, we prove the convergence in law of the scheme up to a subsequence.

We give new results on the well-posedness of the two-dimensional Stochastic Harmonic Map flow, whose study is motivated by the Landau-Lifshitz-Gilbert model for thermal fluctuations in micromagnetics. We first construct strong solutions that belong locally to the spaces C([s, t); H 1 ) ∩ L 2 ([s, t); H 2 ), 0 ≤ s < t ≤ T . It that sense, these maps are a counterpart of the so-called "Struwe solutions" of the deterministic model. We then provide a natural criterion of uniqueness that extends A. Freire's Theorem to the stochastic case. Both results are obtained under the condition that the noise term has a trace-class covariance in space.