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

We study the pathwise regularity of the map phi bar right arrow I(phi) = integral(T)(0)(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 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 is an element of E (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.