Spectral gap for spherically symmetric log-concave probability measures, & beyond
主 题: Spectral gap for spherically symmetric log-concave probability measures, & beyond
报告人: 马宇韬副教授 (北京师范大学)
时 间: 2014-09-29 15:00-16:00
地 点: 英国威廉希尔公司理科一号楼1303（概率论系列报告）
Let $\mu$ be a probability measure on $\mathbb{R} ^n$ ($n \geq 2$) with Lebesgue density proportional to $e^{-V (\Vert x\Vert )}$, where $V : \mathbb{R} _+ \to \mathbb{R}$ is a smooth convex potential. We show that the associated spectral gap in $L^2 (\mu)$ lies between $(n-1) / \int_{\mathbb{R} ^n} \Vert x\Vert ^2 \mu(dx)$ and $n / \int_{\mathbb{R} ^n} \Vert x\Vert ^2 \mu(dx)$, 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 dynamics in the energy, leading to weighted Poincar\'e inequalities. All our results are illustrated by some classical and less classical examples.
This is Joint work with M. Bonnefont, A., Joulin.
