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MiS Preprint

Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape

Georg Menz and André Schlichting


We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian $H:\mathbb{R}^n\to \mathbb{R}$ in the regime of low temperature $\varepsilon$. We proof the Eyring-Kramers formula for the optimal constant in the Poincaré (PI) and logarithmic Sobolev inequality (LSI) for the associated generator $L= \varepsilon \triangle - \nabla H \cdot \nabla$ 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 $\varepsilon$. 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.

MSC Codes:
60J60, 35P15, 49R05
Diffusion process, Eyring-Kramers formula, Kramers Law, metastability, Poincaré inequality, spectral gap, Logarithmic Sobolev inequality, weighted transport distance

Related publications

2014 Repository Open Access
Georg Menz and André Schlichting

Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape

In: The annals of probability, 42 (2014) 5, pp. 1809-1884