The Wave equation in a double well potential, with random invariant initial data is expected to exhibit metastable behaviour: as the temperature approaches 0, the solution spends exponentially long times near the wells and the behaviour of the expected transition times is to be given by the Eyring Kramers Law. The amount of crossings between the wells is bounded by the amount of crossings of a hyperplane between them. We establish an Eyring-Kramers formula for the hyperplane crossings in 2 and 3 dimensions, building on work of Vanden-Eijnden and Newhall in 1 dimension.
This is joint work with Petri Laarne (University of Helsinki).