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

Michel Bonnefont (1) , Aldéric Joulin (2) , Yutao Ma (3)

1 IMB - Institut de Mathématiques de Bordeaux
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 BNU - Beijing Normal University

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.

Mots clés: Log- concave probability measure; Poincaré-type inequalities; Diffusion operator; Spectral gap

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

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