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

@article{Menz2014PoincarAL,
title={Poincar{\'e} and logarithmic Sobolev inequalities by decomposition of the energy landscape},
author={G. Menz and Andr{\'e} Schlichting},
journal={Annals of Probability},
year={2014},
volume={42},
pages={1809-1884}
}
• Published 2014
• Mathematics, Physics
• Annals of Probability
• 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 Poincar\'{e} (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… CONTINUE READING

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