Uniform Large Deviations, Barrier Identity and the Néel--Arrhenius Law for the One-Dimensional Stochastic Landau--Lifshitz--Gilbert Equation
Abstract
We study the one-dimensional stochastic Landau-Lifshitz-Gilbert equation with a regularised, multiplicative, degenerate trace-class noise of strength sqrt(eps), the regularisation being Q_delta^{1/2} = (I + delta A^{beta/2})^{-1}, A = -d^2/dx^2, beta > 3/2, and determine the exponential rate of thermally induced magnetisation reversal. (i) Uniform large deviations at fixed delta: for every delta > 0 the solution satisfies a large deviation principle on C([0,T];H), with a good rate function, uniformly in the initial datum over a V-bounded set with compact H-closure; no non-degeneracy of the noise is assumed. (ii) Removal of the regularisation: as delta tends to 0 the noise tends to space-time white noise. With U_delta the state-space quasi-potential and U_0 that of the limiting control problem (Q_0^{1/2} = I), U_delta converges to U_0 in the Gamma-sense, and the barrier is delta-independent: the infimum of U_delta over the boundary of D equals that of U_0 and equals c_* times the infimum of the micromagnetic energy E over that boundary, c_* = 2 lambda_2/(lambda_1^2 + lambda_2^2), under two structural assumptions only and with no geometric or transversality input. Pointwise convergence fails, so the Gamma-limit is the sharp form of the statement. (iii) Neel-Arrhenius law: combining (i) and (ii) with the classical exit-time theorem, the escape time tau from D = {E < r} satisfies both levels of the law - for every fixed delta > 0, eps log E[tau] converges, uniformly on compact subsets of the initial set, to the barrier, and lim_{delta->0} lim_{eps->0} eps log E[tau] = c_* times the infimum of E over the boundary of D. The Arrhenius exponent is therefore stable under the regularisation of the noise and is determined by the micromagnetic energy alone.
Related
- The stochastic Landau-Lifshitz-Baryakhtar equation in critical spaces
- Small-Time Asymptotic Behavior of the Stochastic Landau--Lifshitz--Baryakhtar Equation
- Rate of convergence of a fully discrete structure-preserving midpoint scheme for the stochastic Landau--Lifshitz--Gilbert equation
- Well-posedness of the Stochastic Landau--Lifshitz--Baryakhtar Equation in ℝd
- Continuation and threshold global strong solutions for the two-dimensional Landau--Lifshitz--Gilbert equation
BibTeX
@misc{brzezniak2026uniform,
title = {{Uniform Large Deviations, Barrier Identity and the Néel--Arrhenius Law for the One-Dimensional Stochastic Landau--Lifshitz--Gilbert Equation}},
author = {Zdzisław Brzeźniak and Fan Gu and Liang Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34823},
eprint = {2610.34823},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34823},
}
RIS
TY - UNPB
TI - Uniform Large Deviations, Barrier Identity and the Néel--Arrhenius Law for the One-Dimensional Stochastic Landau--Lifshitz--Gilbert Equation
AU - Zdzisław Brzeźniak
AU - Fan Gu
AU - Liang Li
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34823
UR - https://arxiv.org/abs/2610.34823
AB - We study the one-dimensional stochastic Landau-Lifshitz-Gilbert equation with a regularised, multiplicative, degenerate trace-class noise of strength sqrt(eps), the regularisation being Q_delta^{1/2} = (I + delta A^{beta/2})^{-1}, A = -d^2/dx^2, beta > 3/2, and determine the exponential rate of thermally induced magnetisation reversal. (i) Uniform large deviations at fixed delta: for every delta > 0 the solution satisfies a large deviation principle on C([0,T];H), with a good rate function, uniformly in the initial datum over a V-bounded set with compact H-closure; no non-degeneracy of the noise is assumed. (ii) Removal of the regularisation: as delta tends to 0 the noise tends to space-time white noise. With U_delta the state-space quasi-potential and U_0 that of the limiting control problem (Q_0^{1/2} = I), U_delta converges to U_0 in the Gamma-sense, and the barrier is delta-independent: the infimum of U_delta over the boundary of D equals that of U_0 and equals c_* times the infimum of the micromagnetic energy E over that boundary, c_* = 2 lambda_2/(lambda_1^2 + lambda_2^2), under two structural assumptions only and with no geometric or transversality input. Pointwise convergence fails, so the Gamma-limit is the sharp form of the statement. (iii) Neel-Arrhenius law: combining (i) and (ii) with the classical exit-time theorem, the escape time tau from D = {E < r} satisfies both levels of the law - for every fixed delta > 0, eps log E[tau] converges, uniformly on compact subsets of the initial set, to the barrier, and lim_{delta->0} lim_{eps->0} eps log E[tau] = c_* times the infimum of E over the boundary of D. The Arrhenius exponent is therefore stable under the regularisation of the noise and is determined by the micromagnetic energy alone.
ER -