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

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.

Nous étudions la régularité trajectorielle de l'opérateur φI(φ)=0Tφ(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.

DOI: 10.1214/08-AIHP174
Classification: 76M35,  60H05,  60H30,  60G18,  60G15,  60G60,  76F55
Keywords: pathwise stochastic integrals, currents, forward and symmetric integrals, fractional brownian motion, vortex filaments
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     publisher = {Gauthier-Villars},
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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, Volume 45 (2009) no. 2, pp. 545-576. doi : 10.1214/08-AIHP174. http://www.numdam.org/articles/10.1214/08-AIHP174/

[1] E. Alos, J. A. Leon and D. Nualart. Stochastic Stratonovich calculus for fractional Brownian motion with Hurst parameter less than 1/2. Taiwanese J. Math. 5 (2001) 609-632. | MR | Zbl

[2] K. D. Elworthy, X.-M. Li and M. Yor. The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes. Probab. Theory Related Fields 115 (1999) 325-355. | MR | Zbl

[3] P. Embrechts and P. M. Maejima. Selfsimilar Processes. Princeton University Press, Princeton, NJ, 2002. | MR | Zbl

[4] F. Flandoli. On a probabilistic description of small scale structures in 3D fluids. Annal. Inst. H. Poincaré Probab. Statist. 38 (2002) 207-228. | Numdam | MR | Zbl

[5] F. Flandoli and M. Gubinelli. The Gibbs ensemble of a vortex filament. Probab. Theory Related Fields 122 (2002) 317-340. | MR | Zbl

[6] F. Flandoli and M. Gubinelli. Statistics of a vortex filament model. Electron. J. Prob. 10 (2005) 865-900. | MR | Zbl

[7] F. Flandoli and M. Gubinelli. Random Currents and Probabilistic Models of Vortex Filaments. Birkäuser, Basel, 2004. | MR | Zbl

[8] F. Flandoli and I. Minelli. Probabilistic models of vortex filaments. Czechoslovak Math. J. 51 (2001) 713-731. | MR | Zbl

[9] F. Flandoli, M. Giaquinta, M. Gubinelli and V. M. Tortorelli. Stochastic currents. Stochastic Process. Appl. 115 (2005) 1583-1601. | MR | Zbl

[10] J.-F. Le Gall. Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan. Séminaire de probabilités, XIX, 1983/84 314-331. Lecture Notes in Math. 1123. Springer, Berlin, 1985. | Numdam | MR | Zbl

[11] M. Gradinaru, F. Russo and P. Vallois. Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index H≥¼. Ann. Probab. 31 (2003) 1772-1820. | MR | Zbl

[12] M. Gradinaru, I. Nourdin, F. Russo and P. Vallois. m-order integrals and generalized Itô's formula: the case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 781-806. | Numdam | MR | Zbl

[13] M. Gradinaru and I. Nourdin. Approximation at first and second order of m-order integrals of the fractional Brownian motion and of certain semimartingales. Electron. J. Probab. 8 (2003) 26 pp. | MR | Zbl

[14] M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86-140. | MR | Zbl

[15] T. J. Lyons and Z. Qian. System Control and Rough Paths. Oxford University Press, 2002. | MR | Zbl

[16] T. J. Lyons. Differential equations driven by rough signals. Revista Math. Iberoamericana 14 (1998) 215-310. | MR | Zbl

[17] D. Nualart, C. Rovira and S. Tindel. Probabilistic models for vortex filaments based on fractional Brownian motion. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95 (2001) 213-218. | MR | Zbl

[18] A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, 1983. | MR | Zbl

[19] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer-Verlag, Berlin, 1999. | MR | Zbl

[20] F. Russo and P. Vallois. Stochastic calculus with respect to continuous finite quadratic variation processes. Stochastics Stochastics Rep. 70 (2000) 1-40. | MR | Zbl

[21] F. Russo and P. Vallois. Elements of stochastic calculus via regularization. Séminaire de Probabilités XL 147-186. C. Donati-Martin, M. Emery, A. Rouault and C. Stricker (Eds). Lecture Notes in Math. 1899. Springer, Berlin, Heidelberg, 2007. | MR | Zbl

[22] H. Triebel. Interpolation Theory, Function Spaces, Differential Operators. North Holland, Amsterdam, 1978. | MR | Zbl

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