Abstract
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 property. 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.
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08 August 2017
An erratum to this article has been published.
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Communicated by P. Constantin
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 Brzeźniak 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.
An erratum to this article is available at https://doi.org/10.1007/s00205-017-1139-7.
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Brzeźniak, Z., Goldys, B. & Jegaraj, T. Large Deviations and Transitions Between Equilibria for Stochastic Landau–Lifshitz–Gilbert Equation. Arch Rational Mech Anal 226, 497–558 (2017). https://doi.org/10.1007/s00205-017-1117-0
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DOI: https://doi.org/10.1007/s00205-017-1117-0