Dynamic large deviations: nucleation and growth in phase transitions and avalanches in random hamiltonian systems

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Stochastic perturbations allow for large deviations from the deterministic evolutions, as for example the escape from (deterministically) stationary states. We investigate dynamic large deviations in two different contexts: mesoscale models for the evolution of phase boundaries in multi-phase materials, and overdamped Langevin dynamics with a 'wiggly' potential as a possible model for avalanches.The focus of the first problem is on the existence and qualitative properties of good rate functionals and the characterisation of minimisers under constraints on the macroscopic evolution. A particular example is the switching (tunneling) between stationary states of the deterministic evolution.The aim of the second problem is to identify a stochastic dynamic in a suitable large deviations scaling of a simplified random Hamiltonian system with a potential that has a highly oscillating ('wiggly') part. The main question is whether this can serve as a model for avalanches.