Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model

Franco Flandoli, Massimiliano Gubinelli, and Francesco Russo

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Abstract

We study the pathwise regularity of the map

φI(φ)=0Tφ(Xt), dXt〉,

where φ is a vector function on ℝd belonging to some Banach space V, X is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A continuous version of this map, seen as a random element of the topological dual of V will be called stochastic current. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture that those are also necessary. Next we verify the sufficient conditions when the process X is a d-dimensional fractional Brownian motion (fBm); we identify regularity in Sobolev spaces for fBm with Hurst index H∈(1/4, 1). Next we provide some results about general Sobolev regularity of currents when W is a standard Wiener process. Finally we discuss applications to a model of random vortex filaments in turbulent fluids.

Résumé

Nous étudions la régularité trajectorielle de l’opérateur

φI(φ)=0Tφ(Xt), dXt〉,

φ est une fonction vectorielle à valeurs dans ℝd appartenant à un certain espace de Banach V, X est un processus stochastique et l’intégrale est une certaine version d’une intégrale stochastique définie via régularisation. Une version continue d’un tel opérateur, interprétée comme une variable aléatoire à valeurs dans le dual topologique de V sera appelée courant stochastique. Nous donnons des conditions suffisantes pour que le courant se situe dans un certain espace de Sobolev de distributions. De plus nous donnons des arguments qui permettent de conjecturer que ces conditions sont aussi nécessaires. Successivement nous vérifions la validité de ces conditions lorsque le processus X est un mouvement brownien fractionnaire (mbf) d-dimensionnel; en particulier, nous identifions la régularité de Sobolev pour un mbf d’indice de Hurst H∈(1/4, 1). Par suite, nous fournissons quelques résultats sur la régularité générale de Sobolev de courants relative à un mouvement brownien standard. Enfin nous discutons une application à un modèle de filaments de vorticité dans un fluide turbulent.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 45, Number 2 (2009), 545-576.

Dates
First available in Project Euclid: 29 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1241024680

Digital Object Identifier
doi:10.1214/08-AIHP174

Mathematical Reviews number (MathSciNet)
MR2521413

Zentralblatt MATH identifier
1171.76019

Subjects
Primary: 76M35: Stochastic analysis 60H05: Stochastic integrals 60H30: Applications of stochastic analysis (to PDE, etc.) 60G18: Self-similar processes 60G15: Gaussian processes 60G60: Random fields 76F55: Statistical turbulence modeling [See also 76M35]

Keywords
Pathwise stochastic integrals Currents Forward and symmetric integrals Fractional Brownian motion Vortex filaments

Citation

Flandoli, Franco; Gubinelli, Massimiliano; Russo, Francesco. On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 2, 545--576. doi:10.1214/08-AIHP174. https://projecteuclid.org/euclid.aihp/1241024680.


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