In another question, I've asked how we can derive a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories.
More concretely, I want to obtain a SDE of type as studied in A Concise Course on Stochastic Partial Differential Equations.
If $t\mapsto\Phi_t$ is the trajectory of a particle, I've assumed that $${\rm d}\Phi_t=u(t,\Phi_t){\rm d}t+\xi(t,\Phi_t){\rm d}W_t\;.\tag 1$$ In an abstract setting, $\Phi$ takes values in a Hilbert space $H$ and $W$ is a cylindrical Wiener process on another Hilbert space $U$.
Using an Itō formula, we obtain $${\rm d}\langle u_t(\Phi_t),x_n\rangle_H=\left[\langle\frac{\partial u}{\partial t}(t,\Phi_t)+{\rm D}u(t,\Phi_t)u(t,\Phi_t)+\frac 12\sum_{k\in\mathbb N}{\rm D}^2(t,\Phi_t)\left(\xi_t(\Phi_t)e_k\right)\left(\xi_t(\Phi_t)e_k\right),x_n\rangle_H\right]{\rm d}t+{\rm D}u(t,\Phi_t)\xi_t(\Phi_t)\;{\rm d}W_t\tag 2$$ where $(x_n)_n$ and $(e_n)_n$ are arbitrary orthonormal bases of $H$ and $U$, respectively.
Choosing $H=\mathbb R^d$ and $x_1,\ldots,x_d$ to be the standard basis, $(2)$ becomes $${\rm d}u^{(i)}(t,\Phi_t)=\left[\frac{\partial u^{(i)}}{\partial t}(t,\Phi_t)+\left(u(t,\Phi_t)\cdot\nabla\right)u^{(i)}(t,\Phi_t)+\sum_{k\in\mathbb N}\nabla^2u^{(i)}(t,\Phi_t)\left(\xi_t(\Phi_t)e_k\cdot\xi_t(\Phi_t)e_k\right)\right]{\rm d}t+\left(\xi_t(\Phi_t)\cdot\nabla\right)u^{(i)}(t,\Phi_t)\;{\rm d}W_t\tag 3$$ where $u^{(i)}$ is the $i$th component of $u$ and $\nabla^2u^{(i)}$ denotes the Hessian of $u^{(i)}$.
How can we recast $(3)$ as an evolution equation in a Hilbert space of functions in space (e.g. $L^2(\Omega)$, $\Omega$ being the domain occupied by the fluid)?
(I don't see any way to obtain such an equation directly, but feel free to tell me that I'm wrong.)
