Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Tube estimates for diffusions under a local strong Hörmander condition

Vlad Bally, Lucia Caramellino, and Paolo Pigato

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Abstract

We study lower and upper bounds for the probability that a diffusion process in $\mathbb{R}^{n}$ remains in a tube around a deterministic skeleton path up to a fixed time. The diffusion coefficients $\sigma_{1},\ldots,\sigma_{d}$ may degenerate, but we assume that they satisfy a strong Hörmander condition involving the first order Lie brackets around the skeleton of interest. The tube is written in terms of a norm which accounts for the non-isotropic structure of the problem: in a small time $\delta$, the diffusion process propagates with speed $\sqrt{\delta}$ in the direction of the diffusion vector fields $\sigma_{j}$ and with speed $\delta$ in the direction of $[\sigma_{i},\sigma_{j}]$. We first prove short-time (non-asymptotic) lower and upper bounds for the density of the diffusion. Then, we prove the tube estimate using a concatenation of these short-time density estimates.

Résumé

On étudie des bornes inférieures et supérieures pour la probabilité qu’un processus de diffusion dans $R^{n}$ reste dans un petit tube autour d’un squelette déterministe jusqu’à un temps fixé. Les coefficients de diffusion $\sigma_{1},\dots,\sigma_{d}$ peuvent dégénérer, mais on suppose qu’ils satisfont à une condition d’Hörmander forte sur les coefficients et leurs crochets de Lie de premier ordre autour du squelette d’intérêt. Le tube est écrit en termes d’une norme qui prend en compte la structure non isotrope du problème: en temps $\delta$ petit, le processus de diffusion se propage avec vitesse $\sqrt{\delta}$ dans la direction des vecteurs de diffusion $\sigma_{j}$ et avec vitesse $\delta$ dans la direction de $[\sigma_{i},\sigma_{j}]$. On prouve d’abord des bornes inférieures et supérieures en temps court (non asymptotiques) pour la densité de la diffusion. Ensuite, on prouve l’estimée de tube en utilisant une concaténation de ces estimées de densité en temps court.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 4 (2019), 2320-2369.

Dates
Received: 11 October 2016
Revised: 12 October 2018
Accepted: 12 November 2018
First available in Project Euclid: 8 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1573203631

Digital Object Identifier
doi:10.1214/18-AIHP950

Mathematical Reviews number (MathSciNet)
MR4029156

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Tube estimates Short-time density estimates Hypoellipticity Strong Hörmander condition Malliavin calculus

Citation

Bally, Vlad; Caramellino, Lucia; Pigato, Paolo. Tube estimates for diffusions under a local strong Hörmander condition. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 4, 2320--2369. doi:10.1214/18-AIHP950. https://projecteuclid.org/euclid.aihp/1573203631


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