2006 # Action minimization and sharp‐interface limits for the stochastic Allen‐Cahn equation

**Abstract:** We study the action minimization problem that is formally associated to phase transformation in the stochastically perturbed Allen-Cahn equation. The sharpinterface limit is related to (but different from) the sharp-interface limits of the related energy functional and deterministic gradient flows. In the sharp-interface limit of the action minimization problem, we find distinct "most likely switching pathways," depending on the relative costs of nucleation and propagation of interfaces. This competition is ca…

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“…For problems with the complexity of micromagnetics or martensitic phase transformation this question remains open. However for the simpler case of a scalar Ginzburg-Landau model there has been some progress [42], [43].…”

confidence: 99%

“…For problems with the complexity of micromagnetics or martensitic phase transformation this question remains open. However for the simpler case of a scalar Ginzburg-Landau model there has been some progress [42], [43].…”

confidence: 99%

“…In two space dimensions the problem has been studied numerically in [24] and analytically in [42], but a complete analysis is still lacking. The anticipated answer is similar to the one-dimensional case, except that (i) if the seeds are points rather than lines then their nucleation cost is negligible; and (ii) if a boundary moves via motion by mean curvature then its propagation cost is negligible.…”

confidence: 99%

“…• The upper bound necessary for the Gamma-convergence was formally proved [17] by the construction of good 'recovery sequences'.…”

confidence: 99%

“…• The lower bound was proved in [17] for sequences (u ε ) ε>0 such that the associated 'energy measures' have equipartitioned energy and single multiplicity as ε → 0.…”

confidence: 99%