respect to $x$. That means the following: the integral (\[int\]) belongs to the Sobolev -Watanabe space $\mathbb{D} ^{-\alpha , 2} $ for every $\alpha >\frac{1}{2}$ and to the Sobolev space $H^{-r}(\mathbb{R}; \mathbb{R})$ for $r> \frac{1}{2}$. We notice the same interesting phenomenon already remark in the one-dimensional case (see [@FT]): the order of regularity of (\[int\]) is the same with respect to $\omega$ and with respect to $x$.

Our paper is structured as follows: Section 2 contains some preliminaries on multiple Wiener-Itô integrals and the divergence integral. In Section 3 we explain how we arrive to the our definition of the 2D -stochastic currents, making links with the stochastic calculus for two-parameter processes. In Sections 4 and 5 we analyze the 2D-stochastic current over the Brownian sheet in Watanabe spaces and Sobolev spaces respectively. Finally in Section 6 we treat the 2D -stochastic current over the $d$ -dimensional Brownian sheet.

Malliavin calculus, chaos expansion and Sobolev-Watanabe spaces {#prel}
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Here we describe the elements from stochastic analysis that we will need in the paper. Consider ${\mathcal{H}}$ a real separable Hilbert space and $(B (\varphi), \varphi\in{\mathcal{H}})$ an isonormal Gaussian process on a probability space $(\Omega, {\cal{A}}, P)$, that is a centered Gaussian 