1D -currents, the regularity of (\[xi\]) is the same with respect to the space variable $x$ and with respect to the randomness variable $\omega$.

Convergence of the Fourier transforms
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Here we study the convergence of the sequence $T_{N,M}$ (\[TNM\]) to the 2D-stochastic current (\[xi\]) in the Sobolev space. Precisely, we have the following:

Let $T_{N,M}$ be given by (\[TNM\]). Then as $M,N\to \infty$, the sequence $T_{N,M}$ converges to $\xi$ in $L^{2}(\Omega; H^{-r}(\mathbb{R}; \mathbb{R}))$ for every $r>\frac{1}{2}$.

[**Proof:** ]{} From (\[TNM\]) it is immediate that $$T_{N,M}(\hat{x}) =\sum_{i=0}^{N-1} \sum_{j=0}^{M-1} e^{_ixW_{s_{i}, t_{j}}} \left( W_{s_{i+1}, t_{j}} -  W_{s_{i}, t_{j}}\right) \left(  W_{s_{i}, t_{j+1}}- W_{s_{i}, t_{j}}\right)$$ where $T_{N,M}(\hat{x})$ denotes the Fourier transform of $x\to T_{N,M}(x).$ We also know from Proposition \[pp1\] that $$\xi (\hat{x})=\delta ^{(2)}_{(u_{1},v_{1}), (u_{2}, v_{2}) }\left( e ^{-xW_{u_{1}, v_{2}} } 1_{[0,u_{1}]} (u_{2} ) 1_{[0,v_{2} ] } (v_{1} ) \right).$$ Then the $L^{2}(\Omega; H^{-r}(\mathbb{R}; \mathbb{R}))$-norm of the difference $T_{N,M}-\xi$ can be computed as follows $$\begin{aligned}
\mathbf{E} \left| T_{N,M} -\xi \right| ^{2} _{H^{-r} (\mathbb{R}; \mathbb{R})} =\mathbf{E}\int_{\mathbb{R} } \left| T_{N,M}(\hat{x}) -\xi(\hat{x} ) \right| ^{2} (1+x^{2}) ^{-r} dx.\end{aligned}$$ Let us first compute $$\begin{aligned}
&& \mathbf{E} \left| T_{N,M}(\hat{x}) -\xi(\hat{x} ) \right| ^{2}\\
&=&\mathbf{E} \left|  \sum_{i=0}^{N-1} \sum_{j=0}^{M-1} \cos (-xW_{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}, 