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We consider a drift-diffusion process on a smooth potential landscape

with small noise. We give a new proof of the Eyring-Kramers formula

which asymptotically characterizes the spectral gap of the generator of

the diffusion. The proof is based on a refinement of the two-scale

approach introduced by Grunewald, Otto, Villani, and Westdickenberg and

of the mean-difference estimate introduced by Chafai and Malrieu. The

new proof exploits the idea that the process has two natural

time-scales: a fast time-scale resulting from the fast convergence to a

metastable state, and a slow time-scale resulting from exponentially

long waiting times of jumps between metastable states. A nice feature

of the argument is that it can be used to deduce an asymptotic formula

for the log-Sobolev constant, which was previously unknown.