Selection of a stochastic Landau-Lifshitz equation and the stochastic persistence problem

In this article, we study the persistence of properties of a given classical deterministic differential equation under a stochastic perturbation of two distinct forms: external and internal. The first case corresponds to add a noise term to a given equation using the framework of Ito or Stratonovich stochastic differential equations. The second case corresponds to consider parameter dependent differential equations and to add a stochastic dynamics on the parameters using the framework of random ordinary differential equations. Our main concerns for the preservation of properties are stability/instability of equilibrium points and symplectic/Poisson Hamiltonian structures. We formulate persistence theorem in these two cases and prove that the cases of external and internal stochastic perturbations are drastically different. We then apply our results to develop a stochastic version of the Landau-Lifshitz equation. We discuss, in particular, previous results obtained by Etore et al. [J. Differ. Equations 257, 2115-2135 (2014)] and we finally propose a new family of stochastic Landau-Lifshitz equations.