## The Annals of Probability

### Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape

#### Abstract

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.

#### Article information

Source
Ann. Probab. Volume 42, Number 5 (2014), 1809-1884.

Dates
First available in Project Euclid: 29 August 2014

https://projecteuclid.org/euclid.aop/1409319468

Digital Object Identifier
doi:10.1214/14-AOP908

Mathematical Reviews number (MathSciNet)
MR3262493

#### Citation

Menz, Georg; Schlichting, André. Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape. Ann. Probab. 42 (2014), no. 5, 1809--1884. doi:10.1214/14-AOP908. https://projecteuclid.org/euclid.aop/1409319468.

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