^{\alpha } m!c_{m}^{2} \int_{0}^{1} \int_{0}^{1} dudv (uv) ^{\frac{1}{2}}\\
&=& \frac{1}{2\pi}\frac{4}{9}\sum_{m\geq 0} (1+m) ^{\alpha } m!c_{m}^{2}\end{aligned}$$ and this is finite for every $\alpha <\frac{-1}{2}$ because, by Stirling’s formula, $m!c_{m}^{2}$ (this has been already used in [@NV], [@Ed], [@CNT]) behaves as $n^{-\frac{1}{2}}$.

We remark that the regularity of the functional $\xi (x)$ in the Watanabe sense is the same as in the one-parameter case (see [@FT]). This implies in particular that it is not a random variable, but a distribution.

Note that the integration interval for the stochastic integral $dW$ in the definition of $\xi(x)$ is the entire interval $[0,1]$. In the one-parameter case (see [@FT]), we need to consider the quantity $\xi(x)= \int_{a} ^{1} \Delta (x-B_{s})dB_{s}$ with $a>0$ in order to obtain that $\xi (x) \in \mathbb{D}^{\alpha , 2}$ with $\alpha <\frac{-1}{2}$.

Approximation of 2D-stochastic current in Watanabe spaces
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0.3cm Let us introduce the Riemann sum $$\label{TNM}
T_{N,M} (x)= \sum_{i=0} ^{N-1} \sum_{j=0} ^{M-1} \Delta (x-W_{s_{i}, t_{j}} ) \left( W_{s_{i+1}, t_{j}}-W_{s_{i}, t_{j}}\right) \left( W_{s_{i}, t_{j+1}}-W_{s_{i}, t_{j}}\right).$$

Following the lines of Theorem \[th1\], we can prove that $T_{N,M}(x)$ belongs to the Sobolev-Watanabe space $\mathbb{D} ^{\alpha, 2 }$ for every $\alpha <\frac{-1}{2}$. This is also a consequence of the proof of Theorem \[tt2\] 