1202.1510 (Georg Menz et al.)

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

Georg Menz, André Schlichting

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.

View original: http://arxiv.org/abs/1202.1510