Distributive of the classical Liouville operator

1. Oct 22, 2009

bubbloy

I'm trying to derive fokker-planck equations using the Liouville operator for classical systems. part of it relies on a property i've seen in many places, that applying the Liouville operator $$e^{Lt}$$ to a product of two functions $$A(\textbf{X})B(\textbf{X})$$ is the same as applying the Liouville operator to both of them individually, i.e.:

$$e^{Lt}(A(\textbf{X})\cdot B(\textbf{X})) = (e^{Lt}A(\textbf{X}))\cdot (e^{Lt}B(\textbf{X}))$$

this seems to make sense physically but does anyone know a proof? it seems to be referenced everywhere.