A stochastic invariantization method for Itô stochastic perturbations of differential equations

Abstract

In general, adding a stochastic perturbation to a differential equation possessing an invariant manifold destroys the invariance as far as the Itô formalism is used. In this article, we propose an invariantization method for perturbations in the Itô case which can be used to restore invariance. We then apply our results to develop a stochastic version of the Landau-Lifshitz equation. We discuss in particular previous results obtained by Etore and al. in [P. Étoré, S.Labbé , J. Lelong, Long time behaviour of a stochastic nanoparticle, J. Differential Equations 257 (2014), 2115-2135].

15 pages