Let $X$ be a vector field on a manifold $M$ that induces a complete flow $\Theta_t$. Then the operator family $\Theta_t^*$, $$\Theta_t^*u(x) = u(\Theta_t(x))$$ is a strongly continuous semigroup of operators on the space $C_0(M)$, and also on the spaces $L^p(M)$, $1 \leq p < \infty$ under some extra conditions (e.g. if $M$ is compact).
The infinitesimal generator is obviously the vector field $X$ as a differential operator of order $1$, on some domain that includes the smooth compactly supported functions.
But what is the domain of the infinitesimal generator?