Digital Object Identiﬁer (DOI) 10.1007/s00205-017-1117-0
Arch. Rational Mech. Anal. 226 (2017) 497–558
Large Deviations and Transitions Between
Equilibria for Stochastic
aw Brze´zniak, Ben Goldys
& Terence Jegaraj
Communicated by P. Constantin
We study a stochastic Landau–Lifshitz equation on a bounded interval and
with ﬁnite dimensional noise. We ﬁrst show that there exists a pathwise unique
solution to this equation and that this solution enjoys the maximal regularity prop-
erty. Next, we prove the large deviations principle for the small noise asymptotic
of solutions using the weak convergence method. An essential ingredient of the
proof is the compactness, or weak to strong continuity, 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. We also 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 from ferromagnetic nanowires to
the fabrication of magnetic memories.
The original version of this article was revised: The presentation of Eqs. 4.6, 6.34, 6.35 and
6.49 has been corrected.
This work is dedicated to memory of our colleague and collaborator, Terence Jegaraj, who
tragically passed away after this work was submitted. We will remember his dedication to
and enthusiasm for mathematics.
The work of Zdzisław Brze´zniak and of Ben Goldys was partially supported by the
ARC Discovery Grant DP120101886. The research on which we report in this paper was
started at the Newton Institute for Mathematical Sciences in Cambridge (UK) during the
program “Stochastic Partial Differential Equations”. The INI support and excellent working
conditions are gratefully acknowledged by all three authors. The ﬁrst named author wishes to
thank Clare Hall (Cambridge) and the School of Mathematics, UNSW, Sydney for hospitality.