process $X$ as defined in the previous section. We also denoted by $$\label{a1}
a_{n}^{x}(s)=  R(s)
^{-\frac{n}{2}}p_{R(s)}(x) H_{n} \left(
\frac{x}{R(s)^{\frac{1}{2}}}\right)$$ for every $s\in S, x\in \mathbb{R}, n\geq 0$.

For $X=W= (W_{s,t}) _{s,t\in [0,1]}$, the Wiener sheet, we have $$\label{deltasheet}
\Delta (x-W_{u_{1}, v_{2}}) = \sum_{m\geq 0} a_{m}^{x} (u_{1}, v_{2}) I_{m} \left(   1_{[0,u_{1}]\times [0, v_{2}]}^{\otimes m} \right)$$ where for every $u_{1}, v_{2} \in [0,1]$, $$\label{amx}
a_{m}^{x} (u_{1}, v_{2}):=(u_{1}v_{2}) ^{-\frac{m}{2}} p_{u_{1}v_{2}} (x) H_{m} \left( \frac{x}{\sqrt{u_{1}v_{2} } } \right) .$$

We will show in the next result that the 2D-stochastic current defined in Definition \[defxi\] is well-defined as a distribution in the Watanabe sense.

\[th1\] For every $x\in \mathbb{R}$ the functional $\xi (x)$ defined in Definition \[defxi\] belongs to the Watanabe space $\mathbb{D} ^{-\alpha ,2} $ for any $\alpha > \frac{1}{2}$.

[**Proof:** ]{} Let $x\in \mathbb{R}$ be fixed throughout the proof. We will compute first the divergence integral $$\delta ^{(2)} _{(u_{1}, v_{1}), (u_{2}, v_{2}) } \left( \Delta (x-W_{u_{1}, v_{2}}) 1_{[0, u_{1}] }(u_{2}) 1_{[0, v_{2}]} (v_{1})\right)$$ with respect to the variables $(u_{1}, v_{1})$ and $(u_{2}, v_{2})$ belonging to $[0,1] ^{2}$. Using (\[deltasheet\]) and the chaotic form of the divergence integral (\[sko\]), we will need to symmetrize the function $$\begin{aligned}
&&\left( (x_{1}, y_{1}), (x_{2}, y_{2}),\ldots , (x_{m+1}, 