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

Franco Flandoli; Massimiliano Gubinelli; Francesco Russo

Annales de l'I.H.P. Probabilités et statistiques (2009)

- Volume: 45, Issue: 2, page 545-576
- ISSN: 0246-0203

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topFlandoli, Franco, Gubinelli, Massimiliano, and 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 45.2 (2009): 545-576. <http://eudml.org/doc/78033>.

@article{Flandoli2009,

abstract = {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.},

author = {Flandoli, Franco, Gubinelli, Massimiliano, Russo, Francesco},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {pathwise stochastic integrals; currents; forward and symmetric integrals; fractional brownian motion; vortex filaments; pathwise stochastic integral; Sobolev space},

language = {eng},

number = {2},

pages = {545-576},

publisher = {Gauthier-Villars},

title = {On the regularity of stochastic currents, fractional brownian motion and applications to a turbulence model},

url = {http://eudml.org/doc/78033},

volume = {45},

year = {2009},

}

TY - JOUR

AU - Flandoli, Franco

AU - Gubinelli, Massimiliano

AU - Russo, Francesco

TI - On the regularity of stochastic currents, fractional brownian motion and applications to a turbulence model

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2009

PB - Gauthier-Villars

VL - 45

IS - 2

SP - 545

EP - 576

AB - 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.

LA - eng

KW - pathwise stochastic integrals; currents; forward and symmetric integrals; fractional brownian motion; vortex filaments; pathwise stochastic integral; Sobolev space

UR - http://eudml.org/doc/78033

ER -

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