Let
- $U$ and $V$ be separable $\mathbb R$-Hilbert spaces
- $\iota:U\to V$ be a Hilbert-Schmidt embedding
- $Q:=\iota\iota^\ast$
- $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(B_n)_{n\in\mathbb N}$ be a family of independent Brownian motions on $(\Omega,\mathcal A,\operatorname P)$ and $$W_n(t):=\sum_{i=1}^dB_i(t)e_i\;\;\;\text{for }n\in\mathbb N\text{ and }t\ge 0\;.$$
It is well-known that $$\left\|W_n(t)-W(t)\right\|_{L^2(\operatorname P,\:V)}\;\xrightarrow{n\to\infty}\;0\;\;\;\text{for all }t\ge 0\tag 1$$ for some $Q$-Wiener process $\left(W\left(t\right)\right)_{t\ge0}$ on $(\Omega,\mathcal A,\operatorname P)$.
Now, imagine that $u=\left(u_1,\ldots,u_d\right):[0,\infty)\times\mathbb R^d\to\mathbb R^d$ is the velocity of a fluid and that $[0,\infty)\ni t\mapsto X_t\in\mathbb R^d$ is the trajectory of a single fluid particle $X_0$.
I want to obtain a SDE for $u$ of type as studied in A Concise Course on Stochastic Partial Differential Equations. As usual, I want to assume that $u$ takes values in $$H:=\overline{\mathfrak D}^{\left\|\;\cdot\;\right\|_{L^2(\Omega,\:\mathbb R^d)}}$$ with $$\mathfrak D:=\left\{\phi\in C_c^\infty(\Omega)^d:\nabla\cdot\phi=0\right\}$$ and $\Omega\subseteq\mathbb R^d$ being open (the "domain" occupied by the fluid).
I want to assume that the trajectory is subject to a random forcing. Having the desired type of SDE for $u$ in mind, I want to choose the forcing to be driven by $W$ (note that we will choose $U$ to be $H$ or a superset of $H$).
So, I want to assume that $${\rm d}X(t)=u(t,X(t)){\rm d}t+\xi(t,X(t)){\rm d}W(t)\;\;\;\text{for all }t\ge 0\tag 2$$ for some $\xi:\Omega\times[0,\infty)\times\mathbb R^d\to\operatorname{HS}(U,\mathbb R^d)$.
By the Itō formula, we obtain $${\rm d}u_i(t,X(t))=\left[\frac{\partial u_i}{\partial t}(t,X(t))+\left(u\left(t,X(t)\right)\cdot\nabla\right)u_i\left(t,X(t)\right)+\frac 12\operatorname{tr}\left[{\xi\left(t,X(t)\right)}^\ast\nabla^2u_i\left(t,X(t)\right)\xi\left(t,X(t)\right)\right]\right]{\rm d}t+\left(\xi\left(t,X(t)\right)\cdot\nabla\right)u_i(t,X_t){\rm d}W(t)\tag 3$$ where $\nabla^2u_i(t,x)$ denotes the Hessian of $u_i$ at $(t,x)\in[0,\infty)\times\mathbb R^d$.
How can we rewrite $(3)$ to obtain a SDE for $u$ of the desired type?
