Abstract: Metastability phenomena show up in the dynamical behavior of a large variety of complex real world systems. From a mathematical point of view the dynamics of such systems may be modeled by means of stochastic differential equations, such as the Langevin equation at small temperature in molecular dynamics. An efficient computation of metastable dynamics over the long time scales of interest is a particularly challenging and important problem for applications.

In this talk I will present some theoretical results providing sharp estimates on key quantities, such as metastable transition times, spectral gaps or exit point distributions. The results concern overdamped Langevin equations in Euclidean domains, discrete space diffusions arising as coarse-grained models in statistical mechanics, and infinite-dimensional systems such as the stochastic Allen-Cahn equation. The proofs are based on a spectral-theoretical approach and employ PDE tools of semiclassical analysis, adapted to the different settings.