Let

 - $d\in\left\{2,3\right\}$
 - $\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
 - $\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\Phi_t(x_0)$ is the position at time $t\ge 0$ of a particle which started at position $x_0\in\mathcal V_0$ at time $0$

Then, the velocity field $u_t:\mathcal V_t\to\mathbb R^d$ at time $t\ge 0$ is defined by $${\rm d}\Phi_t(x_0)=\underbrace{u_t\left(\Phi_t\left(x_0\right)\right)}_{\displaystyle =:v_t(x_0)}{\rm d}t\;\;\;\text{for all }x_0\in\mathcal V_0\;.\tag 1$$ Using $(1)$ and the chain rule, we obtain $$\frac{\partial v}{\partial t}(t,x_0)=\left[\frac{\partial u}{\partial t}+(u\cdot\nabla)u\right]\left(t,\Phi_t\left(x_0\right)\right)\;\;\;\text{for all }t\ge 0\text{ and }x_0\in\mathcal V_0\;.\tag 2$$ By the momentum conservation equation $$\frac{\partial v}{\partial t}(t,x_0)=\underbrace{\left[\nu\Delta u-\nabla p\right]}_{\displaystyle =:f}\left(t,\Phi_t\left(x_0\right)\right)\;\;\;\text{for all }t\ge 0\text{ and }x_0\in\mathcal V_0\;,\tag 3$$ we obtain $$\left[\frac{\partial u}{\partial t}+(u\cdot\nabla)u\right](t,x)=\left[\nu\Delta u-\nabla p\right](t,x)\;\;\;\text{for all }t\ge 0\text{ and }x\in\mathcal V_t\;.\tag 4$$ The crucial point in $(4)$ is that we can replace $\Phi_t(x_0)$ of $(2)$ and $(3)$ by a uniquely determined $x\in\mathcal V_t$, since by definition $\mathcal V_t=\left\{\Phi_t(x_0):x_0\in\mathcal V_0\right\}$. Thus, we can easily recast $(4)$ into an equation $$\left[\frac{\partial u}{\partial t}+(u\cdot\nabla)u\right](t)=\left[\nu\Delta u-\nabla p\right](t)\;\;\;\text{for all }t\ge 0\tag 5$$ in a Hilbert space $H$ of "functions in space", e.g. $H=L^2(\mathcal V;\mathbb R^d)$ where $\mathcal V=\bigcup_{t\ge 0}\mathcal V_t$.

> Now, I want to add a stochastic forcing to $(1)$ and obtain a stochastic PDE similar to $(5)$ using an Itō formula in place of the chain rule in the derivation. I've tried to consider $${\rm d}\Phi_t(x_0)=u_t\left(\Phi_t\left(x_0\right)\right){\rm d}t+\xi_t\left(\Phi_t\left(x_0\right)\right){\rm d}W_t\;\;\;\text{for all }x_0\in\mathcal V_0\;.\tag 6$$ instead of $(1)$. Therefor, let
>
 - $(\Omega,\mathcal A,\operatorname P)$ be a probability space
 - $U$ be a separable Hilbert space
 - $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace
 - $(W_t)_{t\ge 0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$
 - $\xi:\Omega\times[0,\infty)\times\mathbb R^d\to\operatorname{HS}(U_0,\mathbb R^d)$, where $\operatorname{HS}(U_0,\mathbb R^d)$ is the space of Hilbert-Schmidt operators from $U_0:=Q^{1/2}U$ to $\mathbb R^d$
>
> Using the Itō formula (see [Da Prato, Theorem 4.32](https://books.google.de/books?id=H-VSAwAAQBAJ&pg=PA106&lpg=PA106&dq=%22valued+predictable+process+Bochner+integrable%22&source=bl&ots=u8gTivSmkd&sig=86ZpVCJpDd3Snz5vCd5ri7yjMjo&hl=de&sa=X&ved=0ahUKEwiDyPzyoK_MAhWnKJoKHaGIDAwQ6AEIHTAA#v=onepage&q=%22valued%20predictable%20process%20Bochner%20integrable%22&f=true)) we obtain 
>
\begin{equation}
\begin{split}
{\rm d}u^i(t,x_t)&=\left(\xi_t(x_t){\rm d}W_t\cdot\nabla\right)u^i(t,x_t)\\
&+\left[\frac{\partial u^i}{\partial t}(t,x_t)+\left(u_t(x_t)\cdot\nabla\right)u^i(t,x_t)+\operatorname{tr}\left[\nabla^2u^i(t,x_t)\left(\tilde\xi_t(x_t)\right)\left(\tilde\xi_t(x_t)\right)^\ast\right]\right]{\rm d}t
\end{split}\tag 7
\end{equation}
>
where $x_t:=\Phi_t(x_0)$, $u^i$ is the $i$-th component of $u=(u^1,\ldots,u^d)$, $\nabla^2u^i(t,x_t)$ denotes the Hessian of $u^i$ at $(t,x_t)$ and $\tilde\xi_t(x_t):=\xi_t(x_t)Q^{\frac 12}$.

My goal is to obtain a SPDE as considered by [Da Prato](https://books.google.de/books?id=bxkmAwAAQBAJ&pg=PA186&lpg=PA186&dq=%22We+proceed+to+study+nonlinear+equations%22&source=bl&ots=-3epRfRN3x&sig=wh9rHvfO4Fg4t-y4k-bEoXwhD4k&hl=de&sa=X&ved=0ahUKEwjV5q7PlubMAhWDvxQKHRMIDUUQ6AEIHTAA#v=onepage&q=%22We%20proceed%20to%20study%20nonlinear%20equations%22&f=false). I don't know if my approach is the correct approach to obtain a stochastic Navier-Stokes equation. In any case, I need help in rewriting $(7)$ as an equation in a Hilbert space of functions in $x$. Maybe the trace term can be simplified. And maybe we need to rewrite $(7)$ in a coordinate-free form.

I've asked many questions on MSE ([1](https://math.stackexchange.com/questions/1773143/reformulate-a-spde-parameterized-by-space-and-time-as-an-sde-parameterized-by-ti), [2](https://math.stackexchange.com/questions/1761366/can-we-apply-an-it%C5%8D-formula-to-find-an-expression-for-ft-x-t-if-f-is-taki), [3](https://math.stackexchange.com/questions/1774561/recast-the-scalar-spde-du-t%CE%A6-tx-f-t%CE%A6-txdt%E2%88%87-u-t%CE%A6-tx%E2%8B%85%CE%BE-t%CE%A6-txdw-t), [4](https://math.stackexchange.com/questions/1780619/stochastically-perturbed-fluid-flow-map-rm-d%CE%A6-tx-0-u-t%CE%A6-tx-0-rm-dt%CE%BE), [5](https://math.stackexchange.com/questions/1745391/derivation-of-a-stochastic-navier-stokes-equation-with-multiplicative-noise), [6](https://math.stackexchange.com/questions/1732015/rephrase-a-multiparameter-sde-indexed-by-time-and-space-as-an-infinite-dimension)), but didn't find a solution.