I am interested in knowing more about dynamical systems with random parameters of this type:
$$ \partial_t u(t,\omega) = F(u(t,\omega),Z(\omega))$$.
The simplest way to understand what I mean is to start from a deterministic Cauchy problem in $\mathbb{R}^n$, with parameter $Z \in \mathbb{R}$:
$$\dot u(t) = F(u(t),Z), \qquad u(0) = u_0$$
where $F \colon \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n$. Now, I want to consider the case in which $Z$ is a random variable, with probability density $\rho_Z$. This leads to the Cauchy problem
$$ \partial_t u(t,\omega) = F(u(t,\omega),Z(\omega)), \qquad u(0,\omega) = u_0$$
Hence the evolution equation is deterministic, but $u(t)$ is a random variable. This setup, at first, seems different from the usual SDE evolution equation. I would like to know if you have any references where I can learn about that. In particular, I would like to learn how to derive an evolution equation for the law of $u(t)$ (similar to what one would get with a Fokker-Planck equation for SDEs).