Density minoration of a strongly non-degenerated random variable
type: Item
instance of: scholarly article
title: Density minoration of a strongly non-degenerated random variable
zbMATH Open document ID: 1175.60055
author: Paul Malliavin
author: Eulalia Nualart
published in: Journal of Functional Analysis
publication date: 2009-06-30T00:00:00Z
review text: Consider a \(\mathbb{R}^d\)-valued random variable \(F\) defined on the Wiener space; denote by \(D\) the Malliavin derivative and by \(\delta\) the divergence operator on this space. It is well known that if \(F\) is smooth (in the sense of Malliavin's calculus) and if its Malliavin matrix satisfies some non degeneracy condition, then its law has a density. The aim of this article is to find a lower bound for this density. The main tool is the notion of system of covering vector fields of \(F\); this is a variable \((A_1,\ldots,A_d)\) such that \(D_{A_i}(\phi\circ F)=(\partial_i\phi)\circ F\) for any \(i\) and any smooth \(\phi\). A variable of this type can be found by means of the Malliavin matrix of \(F\), but there are other possible choices. The authors prove that if \(\exp(c\sum_i|\delta(A_i)|^\gamma)\) is integrable for some \(\gamma>1\) and \(c>0\), then the density is bounded below by some \(c_\gamma\,\exp(-c_\gamma\|x\|^{\gamma/(\gamma-1)})\). The proof is based on a radial averaging combined with some Riesz transform estimates.
reviewed by: Jean Picard
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 5572183
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: lower bounds for densities
zbMATH Keywords: divergence
zbMATH Keywords: Riesz transforms
MaRDI profile type: Publication
full work available at URL: j.jfa.2008.11.016
OpenAlex ID: W2032581909
cites work: Quasi-invariance of Brownian measures on the group of circle homeomorphisms and infinite-dimensional Riemannian geometry
cites work: Lower bounds for the density of locally elliptic Itô processes
cites work: Non-existence of infinitesimally invariant measures on loop groups
cites work: Potential theory for hyperbolic SPDEs.
cites work: Fundamental solutions for second order subelliptic operators
cites work: Estimates for the density of a nonlinear Landau process
cites work: Lower bounds for densities of uniformly elliptic random variables on Wiener space
cites work: Q4445184
cites work: Q3771355
cites work: Formules de la moyenne, calcul de perturbations et théoremes d'annulation pour les formes harmoniques
cites work: Q4337939
cites work: Stochastic calculus of variations in mathematical finance.
cites work: Exponential divergence estimates and heat kernel tail.
cites work: The Malliavin Calculus and Related Topics
cites work: Fundamental solutions and geometry of the sum of squares of vector fields
DOI: 10.1016/J.JFA.2008.11.016
description: scientific article; zbMATH DE number 5572183