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 .. We proof the Eyring.Kramers formula for the optimal constant in the Poincaré (PI) and logarithmic Sobolev inequality (LSI) for the associated generator L=....H.. of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald et al. [Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 302.351] and of the mean-difference estimate introduced by Chafaï and Malrieu [Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 72.96]. 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 metastable regions corresponding to the local minima scales well in .. 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.