I learned about dynamical flow $\phi(t_0, x_0, t)$ of dynamical system $\dot x(t) = a(x(t))$: it provides the coordinates $x(t)$ at every instant $t$ starting at instant $t_0$ and coordinates $x_0$. It made me wonder the coordinates expected value and standard deviation by addition of, for example, a gaussian stochastic variable to the initial state $x_0$ i.e. $\phi(t_0, t, x_0 + z)$. Could you help me?
Thank you. :)