$ denotes the double Skorohod integrals which as above acts with respect to the variables $(u_{1}, v_{1}), (u_{2}, v_{2}) \in [0,1] ^{2}$.

0.2cm

The above discussion motivate the following definition of the 2D stochastic current.

\[defxi\] Let $(W_{s,t}) _{s,t\in [0,1]} $ be a Wiener sheet and for every $x\in \mathbb{R}$, $s,t\in [0,1]$ consider the functional $\Delta (x-W_{s,t}) $ which is a distribution in the Watanabe sense. We define the stochastic current over the Wiener sheet $W$ as the mapping $$\label{xi}
x\to \xi (x):= \delta ^{(2)} _{(u_{1}, v_{1}), (u_{2}, v_{2}) } \left( \Delta (x-W_{u_{1}, v_{2}}) 1_{[0, u_{1}] }(u_{2}) 1_{[0, v_{2}]} (v_{1})\right).$$

The meaning of this definition will be clarified in the sequel. The term $\Delta (x-W_{u_{1}, v_{2}})$ is a distribution in the Sobolev-Watanabe spaces. In the rest of our paper we will analyze the mapping $\xi$ introduced above from different perspectives: first as a functional of the randomness $\omega$ and then as a function of $x\in \mathbb{R} ^{d}$.

The 2D-stochastic current in Watanabe spaces
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Let us first analyze the function $\xi (x)$, for fixed $x\in \mathbb{R}$, as a distribution in the Watanabe (or Sobolev-Watanabe) sense (see Section \[prel\]). Concretely, we will prove in this section that for every $x$ the 