Long time behaviour of a stochastic nanoparticle

In this article, we are interested in the behaviour of a single ferromagnetic mono-domain particle submitted to an external field with a stochastic perturbation. This model is the first step toward the mathematical understanding of thermal effects on a ferromagnet. In a first part, we present the stochastic model and prove that the associated stochastic differential equation is well defined. The second part is dedicated to the study of the long time behaviour of the magnetic moment and in the third part we prove that the stochastic perturbation induces a non-reversibility phenomenon. Last, we illustrate these results through numerical simulations of our stochastic model.The main results presented in this article are on the one hand the rate of convergence of the magnetization toward the unique stable equilibrium of the deterministic model and on the other hand a sharp estimate of the hysteresis phenomenon induced by the stochastic perturbation (remember that with no perturbation, the magnetic moment remains constant).