Consider a space-time white noise $\xi$ and the heat semi-group $(P_r).$
The following Kolmogorov type criterion allows to construct modifications in Besov Space (Here we have a partition of unity $(\rho_j)_j,$ for a distribution $\Delta_jf:=\mathscr{F}^{-1}\rho_j*f:=K_j*f$):
[![enter image description here][1]][1]
Applying this theorem we get modification for stochastic convolution (for dimensions $d>1$) $X_r(\phi):=\int_0^r\int_{\mathbb{T}^1}(\phi,p(r-q,y-\cdot))\xi(dq,dy),$ and its Wick power: $|\nabla X|^2$ (using Hermite polynomial $H_2$, we can express it in term of second Wiener-Ito integral, using Nelson inequality we deduce that the assumptions of the above theorem are verified).

Let $Y_r:=\int_0^rP_{r-q}(|\nabla X(q)|^2-E[|\nabla X_q|^2])dq$ (Modifications of $Y$ in Besov spaces already exists since we already did the construction for the term inside) and $W(r):=\langle \nabla X_r,\nabla Y_r\rangle.$ (I presume we need to renormalize but probably the constant is $0$)
I was wondering how to do this for $W$? Is it possible to express $W$ as a Wiener-Ito integral? 


  [1]: https://i.sstatic.net/82jtvvVT.jpg