Large deviations and transitions between equilibria for stochastic Landau-Lifshitz-Gilbert equation

Zdzisław Brzeźniak, Ben Goldys and Terence Jegaraj

Abstract

We study a stochastic Landau-Lifshitz equation on a bounded interval and with finite dimensional noise. We first show that there exists a unique pathwise solution to this equation and that solutions enjoy a global maximal regularity property. Next, we prove a large deviations principle for small noise asymptotic of solutions using the weak convergence method. As a byproduct of the proof we obtain compactness of the solution map for a deterministic Landau-Lifschitz equation, when considered as a transformation of external fields. We then apply this large deviations principle to show that small noise can cause magnetisation reversal and also to show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise. The problem is motivated by applications of ferromagnetic nano-wires to the fabrication of magnetic memories.

Keywords: stochastic Landau-Lifschitz equation, strong solutions, maximal regularity, large deviations, Freidlin-Ventzell estimates.

AMS Subject Classification: Primary 35K59; secondary 35R60, 60H15, 82D40.