Open Access
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.


Download Citation

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
Back to Top