# SPECTRAL GAP FOR SPHERICALLY SYMMETRIC LOG-CONCAVE PROBABILITY MEASURES, AND BEYOND

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Abstract : 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.
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Contributor : Aldéric Joulin <>
Submitted on : Tuesday, June 17, 2014 - 5:31:25 PM
Last modification on : Wednesday, June 9, 2021 - 10:00:09 AM
Long-term archiving on: : Wednesday, September 17, 2014 - 11:35:27 AM

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• HAL Id : hal-01009383, version 1
• ARXIV : 1406.4621

### Citation

Michel Bonnefont, Aldéric Joulin, Yutao Ma. SPECTRAL GAP FOR SPHERICALLY SYMMETRIC LOG-CONCAVE PROBABILITY MEASURES, AND BEYOND. Journal of Functional Analysis, Elsevier, 2016, 270. ⟨hal-01009383⟩

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