function $\xi (x)$ belongs to the Watanabe space $\mathbb{D} ^{\alpha, 2} $ for every $\alpha < -\frac{1}{2}$. We also prove that $\xi (x)$ can be approximated, under the norm of the space $\mathbb{D} ^{\alpha, 2}$ by some Riemann sums. In this way, we make the link with the discussion in the previous section.

Regularity of the 2D stochastic current in the Watanabe spaces
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Our method to prove the regularity of the functional (\[xi\]) is based on the Wiener-Itô chaos decomposition. Let $(X_{s})_{s\in S}$ be an isonormal Gaussian process with covariance $$EX_{s}X_{t} :=R(s,t)\hskip0.5cm  s,t \in S.$$ The set $S$ is a subset of $\mathbb{R} ^{N}$. We will use the following decomposition of the delta Dirac function (see Nualart and Vives [@NV], Imkeller et al. [@Imk1], Eddahbi et al. [@Ed]) into orthogonal sum of multiple Wiener-Itô integrals $$\begin{aligned}
\label{deltagen} \Delta (x-X_{s}) &=& \sum _{n\geq 0} R(s)
^{-\frac{n}{2}}p_{R(s)}(x) H_{n} \left(
\frac{x}{R(s)^{\frac{1}{2}}}\right) I_{n}\left( 1_{[0,s]}^{\otimes
n}\right)\\
&: =& \sum _{n\geq 0} a_{n}^{x}(s) I_{n}\left( 1_{[0,s]}^{\otimes
n}\right), \hskip0.5cm x\in \mathbb{R}, s\in S \nonumber\end{aligned}$$ where $R(s):=R(s,s)$, $p_{R(s)}$ is the Gaussian kernel of variance $R(s)$, $H_{n}$ is the Hermite polynomial of degree $n$ and $I_{n}$ represents the multiple Wiener-Itô integral of degree $n$ with respect to the Gaussian 