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 $(\Omega,\mathcal A,\operatorname P)$
Moreover, let
- $d\in\left\{2,3\right\}$
- $\mathcal V_0\subseteq\mathbb R^d$ be a bounded domain
- $u:[0,\infty)\times\mathbb R^d\to\mathbb R^d$ and $\Phi:\Omega\times[0,\infty)\times\mathbb R^d\to\mathbb R^d$ with $${\rm d}\Phi_t(x_0)=u_t\left(\Phi_t\left(x_0\right)\right){\rm d}t+\xi_t\left(\Phi_t\left(x_0\right)\right){\rm d}W_t\;\;\;\text{for all }t\ge 0\text{ and }x_0\in\mathcal V_0\tag 1$$ for some $\xi:\Omega\times[0,\infty)\times\mathbb R^d\to\operatorname{HS}(U_0,\mathbb R^d)$, where $\operatorname{HS}(U_0,\mathbb R^d)$ is the space of Hilbert-Schmidt operators from $U_0:=Q^{1/2}U$ to $\mathbb R^d$
The idea is 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 $0$. By $(1)$, the trajectory of each particle is perturbed by a random forcing. Let's assume that the particle system is closed (i.e. no particle is destroyed and no new particle is created). Then, if $\xi=0$, it's clear that each $\Phi_t$ should be a bijection. If $\xi\ne 0$, I somehow want to preserve that property. In particular, I want to be able to talk about the bounded domain $\mathcal V_t\subseteq\mathbb R^d$ occupied by the particle system at time $t\ge 0$. Is this possible?