Theorem on Variation of probability distribution in a flow

Question

Let's assume I have a dynamical system $H(\mathbf{x},t) \in \mathbb{R}^n$, where the initial phase-space distribution at time $t_0$ in terms of $\mathbf{x}_0$ is gaussian. I define a flow $\mathbf{\Phi}:\mathbf{\Phi}(H(\mathbf{x},t))= G(d)\in\mathbb{R}$; or a coordinate transformation $R: R(H(\mathbf{x},t))=\hat H(\mathbf{\hat x},t) \in \mathbb{R}^n$.
Then is there any theorem that says that my $d$ and $\mathbf{\hat x}$can be represented by the same gaussian distribution.
Or in other words, is there any theorem that states that the distribution of a phase-space will remain the same in a flow and/or transformations.
I checked Liouville's theorem which seems to say something similar, but it doesn't talk about coordinate transformations.