Poincaré and Logarithmic Sobolev Inequalities by Decomposition of the Energy Landscape

We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian H : Rn → R in the regime of low temperature ε. We proof the Eyring-Kramers formula for the optimal constant in the Poincaré (PI) and logarithmic Sobolev inequality (LSI) for the associated generator L = ε∆ − ∇H · ∇ of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Westdickenberg and Villani [GOVW09]; and of the mean-difference estimate introduced by Chafaï and Malrieu [CM10]. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the PI and LSI constant of the diffusion restricted to a basin of attraction of a local minimum scales well in ε. This mimics the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of a mean-difference by a weighted transport distance. It contains the main contribution to the PI and LSI constant, resulting from exponentially long waiting times of jumps between metastable states of the diffusion.