Question Can a flow include a Markov process?
A flow $\{\varphi_t\}$ is a set of maps $\varphi_t: X \to X\ \forall t\in R$ paramterised by $t$ that form a lie algebra on composition, $$ \varphi_t\circ\varphi_s = \varphi_{t+s}\\ \varphi_t^{-1} = \varphi_{-t}\\ \varphi_0 = e $$ where $e$ is the identity operation on $X$.
First thoughts If a flow were to include some Markov process it seems to me that negating the parameter $t$ would not give the inverse map at $t$ since $$ \varphi_t\circ\varphi_{-t} \ne e $$ where the random process causes the difference.
Possible Resolution The first thought as provided has assumed that $t$ has progressed from $t\to t+\delta t$ and provides a notational confusion. If the exact map that was defined at $t$ is inverted then it should contain the same random number that was generated at $t$ else it is not the same map and represents another map in the flow.
I think I have resolved the question to the answer yes but I am looking for a confirmation or a reasoned rebuttal
Reference The Geometric Foundations of Hamiltonian Monte Carlo