Spectral gap estimates at low temperature by decomposition of the energy landscape

Details Spectral gap estimates at low temperature by decomposition of the energy landscape Spectral gap estimates at low temperature by decomposition of the energy landscape

2012-02-07

arxiv.org/abs/1202.1510v1

Mathematics

Probability

Scientific paper

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 small noise $\varepsilon$. We give a new proof of the Eyring-Kramers formula for the spectral gap of 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 Chafa\"{\i} and Malrieu. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the spectral gap of the diffusion restricted to a basin of attraction of a local minimum scales nicely 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 new weighted transport distance. It contains the main contribution of the spectral gap of $L$, resulting from exponential long waiting times of jumps between metastable states of the diffusion.

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