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

Abstract

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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.

How to cite

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Flandoli, 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 -

References

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