While seemingly "obviously true", I fail to verify the following question.
Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \leq T}, P)$ be a complete probability space with normal filtration. Let v: $[0,T] \times \Omega \times \mathbb{R}^d \to \mathbb{R}^d$ be smooth in $(t,x)$ for each $\omega$ and bounded in each $C^N_x$-norm uniformly in $t$ and $\omega$ and assume $v: (t,\omega) \mapsto v(t,\omega)$ is $(\mathcal{F}_t)_t$-adapted, where we consider $v(t,\omega)$ to take values in a suitable space of vector fields on $\mathbb{R}^3$. Consider the transport equation on $[0,T] \times \mathbb{R}^3$ $$du +(v \cdot \nabla)u = 0$$ with initial data $u(0,x) = x$ as a PDE with random coefficients (as $v$ depends on $\omega$) and suppose we have found the $\omega$- unique solution $(t,x) \mapsto u(t,\omega,x)$. Clearly, for each $\omega$, $u(t,\omega,\cdot)$ may be considered in the same space of vector fields chosen above. My question is: Does it follow that $(t,\omega) \mapsto u(t,\omega)$ as a map in this function space is $(\mathcal{F}_t)_t$-adapted, i.e. is the adaptedness of $v$ inherited to $u$?
Intuitively, as $u$ is transported along $v$ and this transport is non-anticipating in time in the sense that the direction of transport at time $t$ does not depend on $v(u)$ for times $u>t$, I would somehow expect a positive answer.