The following is an informal question for which I'd like to (ideally) find a reference. I'm quite a novice in this area but would be happy to find a reference to a theorem along the following lines (if such a thing exists) so that I may apply it in some of my own applied work.
Let $X$ be a separable Hilbert space. Consider a collection of (Borel) probability measures $\mu_t$ on $X$ and a collection of vector fields $v_t: X \to X$ for $t \in [0, T]$, such that the pair satisfy the continuity equation on $[0, T] \times X$ in the distributional sense, i.e. $\partial_t \mu_t + \nabla \cdot (v_t \mu_t) = 0$.
Consider also the flow induced by the vector field $v_t$, i.e. a solution $\phi_t$ to the ODE $$\frac{d}{dt} \phi_t = v_t(\phi_t) \qquad \phi_0 = x$$ with some initial condition $x \in X$, assuming whatever is necessary for such a solution to exist. This flow induces a collection of probability measures given some initial probability measure $\nu_0$ via the pushforward $\nu_t = \phi_{t \textrm{#}} \nu_0$.
In the special case of $X = \mathbb{R}^d$, Proposition 8.1.8 of [Ambrosio, 2005] shows that (under the assumption that $t \mapsto \mu_t$ is narrowly continuous and some additional regularity assumptions on $v_t$) if $(\mu_t, v_t)$ solve the continuity equation, then $\mu_t = (\phi_t)_{\textrm{#}} \mu_0$. That is, in some sense, the flow induced by the vector field $v_t$ generates the path of probability measures.
Question: Does a similar result hold in the general case where $X$ is a separable Hilbert space? I'm happy to place whatever additional assumptions on $\mu_t$ and $v_t$ in order to have such a result hold.
