The Annals of Probability

Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape

Georg Menz and André Schlichting

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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

Permanent link to this document
http://projecteuclid.org/euclid.aop/1409319468

Digital Object Identifier
doi:10.1214/14-AOP908

Mathematical Reviews number (MathSciNet)
MR3262493

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds 49R05: Variational methods for eigenvalues of operators [See also 47A75] (should also be assigned at least one other classification number in Section 49)

Keywords
Diffusion process Eyring–Kramers formula Kramers law metastability Poincaré inequality spectral gap logarithmic Sobolev inequality weighted transport distance

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


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