Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Tube estimates for diffusion processes under a weak Hörmander condition

Paolo Pigato

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Abstract

We consider a diffusion process under a local weak Hörmander condition on the coefficients. We find Gaussian estimates for the density in short time and exponential lower and upper bounds for the probability that the diffusion remains in a small tube around a deterministic trajectory (skeleton path). These bounds depend explicitly on the radius of the tube and on the energy of the skeleton path. We use a norm which reflects the non-isotropic structure of the problem, meaning that the diffusion propagates in $\mathbb{R}^{2}$ with different speeds in the directions $\sigma $ and $[\sigma,b]$. We establish a connection between this norm and the standard control distance.

Résumé

On considère une diffusion dont les coefficients satisfont une condition d’Hörmander faible locale. On obtient des estimées gaussiennes de la densité en temps court et des bornes inférieures et supérieures exponentielles pour la probabilité que la diffusion reste dans un petit tube autour d’une trajectoire déterministe (« squelette »). Ces bornes dépendent explicitement du rayon du tube et de l’énergie du squelette. On utilise une norme qui prend en compte la structure non isotrope du problème, dans le sens où la diffusion se propage dans $\mathbb{R}^{2}$ avec des vitesses différentes dans la direction de $\sigma$ et $[\sigma,b]$. On établit un lien entre cette norme et la distance de contrôle standard.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 299-342.

Dates
Received: 17 December 2014
Revised: 11 October 2016
Accepted: 30 October 2016
First available in Project Euclid: 19 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030830

Digital Object Identifier
doi:10.1214/16-AIHP805

Mathematical Reviews number (MathSciNet)
MR3765891

Zentralblatt MATH identifier
06880056

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Density estimates Tube estimates Hypoellipticity Hörmander condition Malliavin Calculus

Citation

Pigato, Paolo. Tube estimates for diffusion processes under a weak Hörmander condition. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 299--342. doi:10.1214/16-AIHP805. https://projecteuclid.org/euclid.aihp/1519030830


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