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

Russo, Francesco; Flandoli, Franco; Gubinelli, Massimiliano (2009), On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model, Annales de l'I.H.P. Probabilités et Statistiques, 45, 2, p. 545-576. http://dx.doi.org/10.1214/08-AIHP174

Type
Article accepté pour publication ou publié
Date
2009
Journal name
Annales de l'I.H.P. Probabilités et Statistiques
Volume
45
Number
2
Publisher
Institute of Mathematical Statistics
Pages
545-576
Publication identifier
http://dx.doi.org/10.1214/08-AIHP174
Metadata
Show full item record
Author(s)
Russo, Francesco
Flandoli, Franco
Gubinelli, Massimiliano
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.
Subjects / Keywords
Forward and symmetric integrals; Pathwise stochastic integrals; Stochastic Analysis

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