{\cal{H}}\right)$ into $\mathbb{D} ^{\alpha ,
p}$. For adapted integrands, the divergence integral coincides to the classical Itô integral. We will use the notation $$\delta (u) =\int_{0}^{T} u_{s} dB_{s}.$$ Let $u$ be a stochastic process having the chaotic decomposition $u_{s}=\sum _{n\geq 0} I_{n}(f_{n}(\cdot ,s))$ where $f_{n}(\cdot, s)\in {\cal{H}}^{\otimes n}$ for every $s$. One can prove that $u \in {\rm Dom} \ \delta$ if and only if $\tilde f_n
\in {\cal{H}}^{\otimes (n+1)}$ for every $n \geq 0$, and $\sum_{n=0}^{\infty}I_{n+1}(\tilde f_n)$ converges in $L^2(\Omega)$. In this case, $$\label{sko}\delta(u)=\sum_{n=0}^{\infty}I_{n+1}(\tilde f_n) \quad \mbox{and}
\quad \mathbf{E}|\delta(u)|^2=\sum_{n=0}^{\infty}(n+1)! \ \|\tilde
f_n\|_{{\cal{H}}^{\otimes (n+1)}}^{2}.$$ We will also recall the integration by parts formula $$\label{ip}
F\delta (u)= \delta (Fu)+ \langle DF, u\rangle _{{\cal{H}}}$$ which holds for $F\in \mathbb{D} ^{\alpha, 2}, u\in \mathbb{D} ^{\alpha , 2} ({\cal{H}}) $ and $Fu \in \mathbb{D}^{\alpha ,2} ({\cal{H}})$. In the present work we will consider divergence integral with respect to a Brownian motion in $\mathbb{R}^{d}$. Throughout this paper we will denote by $p_{s}(x) $ the Gaussian kernel of variance $s>0$ given by $p_{s}(x)= \frac{1}{\sqrt{2\pi s}} e^{-\frac{x^{2}}{2s}},
x\in \mathbb{R}$ and for $x=(x_{1}, \ldots , x_{d})\in \mathbb{R}^{d}$ by $p_{s} ^{d}(x)= \prod_{i=1}^{d}p_{s}(x_{i}).$

Definition of the 2D- stochastic current {#defint}
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The idea to define a stochastic integral (\[int\]) comes 