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September 2014 Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape
Georg Menz, André Schlichting
Ann. Probab. 42(5): 1809-1884 (September 2014). DOI: 10.1214/14-AOP908


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 $L=\varepsilon \Delta -\nabla H\cdot\nabla$ 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 $\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.


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Georg Menz. André Schlichting. "Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape." Ann. Probab. 42 (5) 1809 - 1884, September 2014.


Published: September 2014
First available in Project Euclid: 29 August 2014

zbMATH: 1327.60156
MathSciNet: MR3262493
Digital Object Identifier: 10.1214/14-AOP908

Primary: 60J60
Secondary: 35P15 , 49R05

Keywords: diffusion process , Eyring–Kramers formula , Kramers law , Logarithmic Sobolev inequality , metastability , Poincaré inequality , spectral gap , weighted transport distance

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 5 • September 2014
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