study of 1D-stochastic currents over the Brownian motion or over the fractional Brownian motion using various interpretation (Itô, Skorohod, Stratonovich etc) for the stochastic integral appearing in the definition of the stochastic current. We will investigate in our paper the 2D-current in the stochastic context using the theory of stochastic integration for two -parameter stochastic processes which has been developed in the eighties (see e.g. [@Dozzi], [@Imk], [@N81]). Somehow, our analysis related geometric aspects of stochastic processes with Malliavin calculus in the spirit of [@Mall].

Let $(W_{s,t} )_{\left(  t,s\right)
\in\left[  0,1\right]  ^{2}}$ be a $\left(  2,d\right)  $-Brownian sheet, with $d>2$. By this we mean a centered Gaussian $d$-dimensional process with $2$-dimensional time parameter $\left(  t,s\right)  \in\left[  0,1\right]  ^{2}$ such that $$Cov\left(  W^{i}_{s,t}, W^{j}_{s^{\prime}, t^{\prime}} \right)  =\delta_{ij}\left(  s\wedge t\right)  \left(  s^{\prime
}\wedge t^{\prime}\right)$$ where $\delta_{ij}$ denotes the Kronecker symbol. Note that the functional (\[map1\]) defines a vector valued distribution $$\int_{0} ^{1} \Delta (x -X_{t})dX_{t}$$ with $\Delta$ the Dirac distribution and the meaning of $ \Delta (x -X_{t})$ will be clarified later. Given (\[map2\]) the correct multiparameter stochastic integrals to be studied for the theory of currents are formally written as $$T_{ij}\left(  x\right)  :=\int_{\left[  0,1\right]  ^{2}}\Delta\left(
x-W_{s,t}  \right)  \partial_{t}W^{i}_{s,t}
\partial_{s}W^{j}_{s,t}$$ with $1\leq 