Spectral gap for spherically symmetric log-concave probability measures, and beyond

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• 作者：Michel Bonnefont^a ; ¹ ; ^{michel.bonnefont@math.u-bordeaux.fr" class="auth_mail" title="E-mail the corresponding author} ; ; Aldé ; ric Joulin^b ; ² ; ^{ajoulin@insa-toulouse.fr" class="auth_mail" title="E-mail the corresponding author} ; ; Yutao Ma^c ; ³ ; ^{mayt@bnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ;
• 关键词：60J60 ; 39B62 ; 37A30 ; 58J50
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 April 2016
• 年：2016
• 卷：270
• 期：7
• 页码：2456-2482
• 全文大小：478 K

文摘

Let μ   be a probability measure on the MathML source">Rⁿ${\mathbb{R}}^n$ (ce4aeda30026f263" title="Click to view the MathML source">n≥2$n\ge 2$) with Lebesgue density proportional to the MathML source">e^{−V(‖x‖)}$e^{-V(\Vert x\Vert )}$, where the MathML source">V:R₊→R$V:{\mathbb{R}}_{+}\to \mathbb{R}$ is a smooth convex potential. We show that the associated spectral gap in the MathML source">L²(μ)$L^2(\mu )$ lies between the MathML source">(n−1)/∫_Rⁿ‖x‖²μ(dx)$(n-1)/{\int }_{{\mathbb{R}}^n}{\Vert x\Vert }^2\mu (dx)$ and the MathML source">n/∫_Rⁿ‖x‖²μ(dx)$n/{\int }_{{\mathbb{R}}^n}{\Vert x\Vert }^2\mu (dx)$, 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 operator in the energy, leading to weighted Poincaré inequalities. All our results are illustrated by some classical and less classical examples.