SPECTRAL GAP FOR SPHERICALLY SYMMETRIC LOG-CONCAVE PROBABILITY MEASURES, AND BEYOND
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Article de revue
Este ítem está publicado en
Journal of Functional Analysis. 2016-04-01, vol. 270
Elsevier
Resumen en inglés
Let $\mu$ be a probability measure on $\rr^n$ ($n \geq 2$) with Lebesgue density proportional to $e^{-V (\Vert x\Vert )}$, where $V : \rr_+ \to \rr$ is a smooth convex potential. We show that the associated spectral gap ...Leer más >
Let $\mu$ be a probability measure on $\rr^n$ ($n \geq 2$) with Lebesgue density proportional to $e^{-V (\Vert x\Vert )}$, where $V : \rr_+ \to \rr$ is a smooth convex potential. We show that the associated spectral gap in $L^2 (\mu)$ lies between $(n-1) / \int_{\rr^n} \Vert x\Vert ^2 \mu(dx)$ and $n / \int_{\rr^n} \Vert x\Vert ^2 \mu(dx)$, improving a well-known two-sided estimate due to Bobkov. Our Markovian approach is remarkably simple and is sufficiently robust to be extended beyond the log-concave case, at the price of potentially modifying the underlying dynamics in the energy, leading to weighted Poincaré inequalities. All our results are illustrated by some classical and less classical examples.< Leer menos
Palabras clave en inglés
Log- concave probability measure
Poincaré-type inequalities
Diffusion operator
Spectral gap
Proyecto ANR
Aux frontières de l'analyse Harmonique - ANR-12-BS01-0013
Géométrie des mesures convexes et discrètes - ANR-11-BS01-0007
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Géométrie des mesures convexes et discrètes - ANR-11-BS01-0007
Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations. - ANR-12-BS01-0019
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