Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape
Georg Menz and André Schlichting
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Submission date: 09. Jul. 2012
MSC-Numbers: 60J60, 35P15, 49R05
Keywords and phrases: Diffusion process, Eyring-Kramers formula, Kramers Law, metastability, Poincaré inequality, spectral gap, Logarithmic Sobolev inequality, weighted transport distance
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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; and of the mean-difference estimate introduced by Chafaï and Malrieu. 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.