  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.
