On the regularity of stochastic currents, fractional brownian motion and applications to a turbulence model
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 545-576.

Nous étudions la régularité trajectorielle de l'opérateur φ↦I(φ)=∫0T〈φ(Xt), dXt〉, où 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.

We study the pathwise regularity of the map φ↦I(φ)=∫0T〈φ(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.

DOI : 10.1214/08-AIHP174
Classification : 76M35, 60H05, 60H30, 60G18, 60G15, 60G60, 76F55
Mots-clés : pathwise stochastic integrals, currents, forward and symmetric integrals, fractional brownian motion, vortex filaments 
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     title = {On the regularity of stochastic currents, fractional brownian motion and applications to a turbulence model},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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Flandoli, Franco; Gubinelli, Massimiliano; Russo, Francesco. On the regularity of stochastic currents, fractional brownian motion and applications to a turbulence model. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 545-576. doi : 10.1214/08-AIHP174. http://www.numdam.org/articles/10.1214/08-AIHP174/

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