### A two-scale proof of the Eyring-Kramers formula

- Series
- Other Talks
- Time
- Tuesday, April 22, 2014 - 11:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Georg Menz – Stanford University

We
consider a diffusion on a potential landscape which is given by a
smooth Hamiltonian in the regime of small noise. We give a new proof of
the Eyring-Kramers
formula for the spectral gap of the associated 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 Eyring-Kramers
formula follows as a simple corollary from two main ingredients : The
first one shows that the Gibbs measure restricted to a domain of
attraction has a "good" Poincaré constant mimicking the fast convergence
of the diffusion to metastable states. The second ingredient is the
estimation of the mean-difference by a new weighted transportation
distance. It contains the main contribution of the spectral gap,
resulting from exponential long waiting times of jumps between
metastable states of the diffusion. This new approach also allows to
derive sharp estimates on the log-Sobolev constant. This is joint work with Andre Schlichting.