Uniform Large Deviations, Barrier Identity and the Néel--Arrhenius Law for the One-Dimensional Stochastic Landau--Lifshitz--Gilbert Equation

Source 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 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.