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Article Dans Une Revue Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques Année : 2009

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

Résumé

We study the pathwise regularity of the map $$ \varphi \mapsto I(\varphi) = \int_0^T \langle \varphi(X_t), dX_t \rangle $$ where $\varphi$ 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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Dates et versions

hal-00134623 , version 1 (03-03-2007)

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Franco Flandoli, Massimiliano Gubinelli, Francesco Russo. On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2009. ⟨hal-00134623⟩
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