Some results on small random perturbations of an infinite dimensional dynamical system

We consider a small random perturbation of a non-linear heat equation with Dirichlet boundary conditions on an interval. The equation can be thought of as a gradient type dynamical system in the space of continuous functions of the interval. It has two stable equilibrium configurations, and several saddle points. We prove that, with probability growing to one in the limit as the strength of the noise goes to zero, the tunnelling between the two stable configurations occurs close to the saddle points with lowest potential. This was suggested by Faris and Jona-Lasinio (1982), who introduced the model. We also prove stability of time averages along a path of the process, in the sense introduced by Cassandro, Galves, Olivieri and Vares (1984), as part of their characterization of metastability for stochastic systems.random perturbations infinite dimensional dynamical systems metastability large deviations