The Allen-Cahn action functional and its sharp interface limit
Matthias Roeger (Max-Planck-Institute Leipzig)
Stochastic perturbations of phase field equations allow
for events that are out of the scope of deterministic models, as for
example the switching between two stable states.
Large deviation theory estimates the probability of such events in
terms of a (deterministic) `action functional'.
Extending previous work by Kohn, Otto, Reznikoff and Vanden-Eijnden we
consider the Allen-Cahn action functional in a particular
scaling that exhibits a competition between nucleation and propagation
costs. Our focus is on a lower bound that is sharp even in the case that
higher multiplicities of the limiting phase interfaces occur. Our
approach is based on techniques from Geometric Measure Theory and uses a
generalized formulation of evolving hypersurfaces similar to that of
Brakke for the mean curvature flow.
(joint work with Luca Mugnai, Leipzig)