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arXiv · 1012.5274

Poincaré inequalities and hitting times

Patrick Cattiaux·Arnaud Guillin·Pierre-André Zitt
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Abstract

Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions are well known. We give here the correspondance (with quantitative results) for reversible diffusion processes. As a consequence, we generalize results of Bobkov in the one dimensional case on the value of the Poincaré constant for logconcave measures to superlinear potentials. Finally, we study various functional inequalities under different hitting times integrability conditions (polynomial, ...). In particular, in the one dimensional case, ultracontractivity is equivalent to a bounded Lyapunov condition.

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Patrick Cattiaux, Arnaud Guillin, Pierre-André Zitt. 2010-12-23. Poincaré inequalities and hitting times. https://arxiv.org/abs/1012.5274

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