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

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Date
2009
Dewey
Probabilités et mathématiques appliquées
Sujet
Forward and symmetric integrals; Pathwise stochastic integrals; Stochastic Analysis
Journal issue
Annales de l'I.H.P. Probabilités et Statistiques
Volume
45
Number
2
Publication date
2009
Article pages
545-576
Publisher
Institute of Mathematical Statistics
DOI
http://dx.doi.org/10.1214/08-AIHP174
URI
https://basepub.dauphine.fr/handle/123456789/2039
Collections
  • CEREMADE : Publications
Metadata
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Author
Russo, Francesco
Flandoli, Franco
Gubinelli, Massimiliano
Type
Article accepté pour publication ou publié
Abstract (EN)
We study the pathwise regularity of the map $$ \phi \mapsto I(\phi) = \int_0^T < \phi(X_t), dX_t>$$ where $\phi$ is a vector function on $\R^d$ 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 \emph{stochastic current} is a continuous version of this map, seen as a random element of the topological dual of $V$. 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 $H \in (1/4,1)$. Next we provide some results about general Sobolev regularity of Brownian currents. Finally we discuss applications to a model of random vortex filaments in turbulent fluids.

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