i\neq j\leq d$. An important question is how to define the above stochastic integral.

At this level, there is no restriction to take $i=1$, $j=2$. Thus we need to define and to study study the mapping $$\label{int}
T\left(  x\right)  :=\int_{\left[  0,1\right]  ^{2}}\Delta\left(  x-W_{s,t}  \right)  \partial_{t}W^{1}_{s,t} \partial_{s}W^{2}_{s,t}.$$ This is actually the aim of this work: to give a meaning to the integral (\[int\]) and to study its regularity in the deterministic Sobolev spaces $H^{-r} (\mathbb{R} ^{d}; \mathbb{R} ^{d})$ as a function of $x$ and in the Sobolev-Watanabe spaces $\mathbb{D}^{\alpha , 2}$ as a function of $\omega$. As we mentioned above, the idea to define the stochastic integral (\[int\]) comes from the stochastic integration theory for two-parameter processes. We will explain it in details in Section \[defint\]. This integral also involves the divergence (Skorohod) integral with respect to the Brownian sheet and multiple Wiener-Itô integrals. Therefore, before defining (\[int\]) we will recall in the next section some elements of the Malliavin calculus for Gaussian processes. We will find the following answer: although the definition of the 2D -stochastic current is substantially different from the 1D-case, the 2D-current inherits the regularity of the one-dimensional case, with respect to $\omega$ as well as with 