Let $d\in\left\{2,3\right\}$ and $\mathcal V_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $x_0\in\mathcal V_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto X_t(x_0)\in\mathcal V_t\tag 1$$ be the movement of $x_0$ over time. Then the speed of movement of $x_0$ along the path $(1)$ is described by $$v_t:\mathcal V_t\to\mathbb R^d\;,\;\;\;X_t(x_0)\mapsto\frac\partial{\partial t}X_t(x_0)\;.\tag 2$$
Now suppose that we want to describe a particle whose motion is perturbed by a random forcing. How can we formulate a stochastic partial differential equation which models this situation?
Intuitively, we should come up with something like $${\rm d}X_t=v_t(X_t){\rm d}t+\xi_t(X_t)\tag 3$$ where $\xi_t(X_t)$ is some random variable.
I've got many problems with $(3)$. In many textbooks the authors consider a Gelfand triple $V\subseteq H\subseteq V^\ast$ and stochastic partial differential equations of the form $${\rm d}Y_t=b_t(Y_t){\rm d}t+\sigma_t(Y_t){\rm d}B_t\tag 4$$ where $B$ is a cylindrical Brownian motion on a separable Hilbert space $U$, $b:[0,\infty)\times V\times\Omega\to V^\ast$ and $\sigma:[0,\infty)\times V\times\Omega\to\left\{\text{Hilbert-Schmidt operators from }U\text{ to }H\right\}$.
Clearly, $X_t$ in $(1)$ is a function $\mathcal V_0\to\mathcal V_t$. Thus, the random variable $X_t$ in $(3)$ should have values in a space of functions $\mathcal V_0\to\mathcal V_t$.
However, I can't find any intuitive choice for $U$ and the Gelfand triple $V\subseteq H\subseteq V^\ast$ for $(3)$. Maybe the Gelfand triple is just a tool used to show some existence/uniqueness results of $(4)$ and I only need to worry about $U$ and $H$ at the modeling stage. Nevertheless, I still don't know how I should choose $U$ and $H$.
Remark: Maybe it's helpful to know, that I want to derive a stochastic Navier-Stokes question from $(3)$.