Florent Barret (Ecole Polytechnique)

Title: Metastability and transition times for a SPDE perturbed by space-time white noise in dimension 1 in space.

Abstract:

We deal with the metastable behavior of the solution of a SPDE parabolic and semilinear, the noise comes from a space-time white noise. This equation could also be seen as the stochastic perturbation of a gradient flow in infinite dimension. We are interested in the asymptotics of the transition times between two minima of our action functional as the intensity of the noise goes to 0. Previous works (Jona-Lasinio, Faris...) showed that as in the finite dimensional case (Freidlin-Wentzell theory) the transition time is asymptotically an exponential random variable, logarithmic asymptotics of the expected time are also known.
We prove in our settings, exact asymptotics of the transition time through a finite difference approximation of our equation (so-called Eyring-Kramers Formula). We apply finite dimensional estimate (Bovier, Eckhoff, Gayrard, Klein) to the system obtained and prove that we can control them uniformly in the dimension (which equals the number of discretization points).

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