We consider the Allen-Cahn equation on the one-dimensional torus, perturbed by a small spacetime white noise. The deterministic equation is a nonlinear PDE, which can be seen as a gradient flow with respect to a double-well energy. If a small noise is added, the typical picture of a metastable dynamics emerges: the system quickly reaches a local equilibrium in one of the two wells; this state endures for an exponentially long time, until a sufficiently large stochastic fluctuation enables the system to overcome the energetic barrier separating the two wells. This behavior produces a slowdown in the relaxation to the equilibrium measure, reflected e.g. by an exponentially small spectral gap. In th alk I will present a technique which provides a formula for the precise asymptotic behavior of the spectral gap, showing that the prefactor is given by a suitable Fredholm determinant. The formula shows that the gap behaves like twice the inverse of the metastable transition time from one well to the other.

IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria www.ist.ac.at © IST 2020