These types of questions are treated in great generality in the book
Rogers-Williams: Diffusions, Markov processes and martingales, Volume 1
One of the great results is due to Dynkin. Let $(X_t)_{t \ge 0}$ be a continuous Markov process whose semigroup is Feller-Dynkin (that is restricts to a strongly continuous semigroup on continuous functions vanishing at $\infty$). Then if the generator of $X$ contains the space of smooth and compactly supported functions, this generator is necessarily a second-order semi-elliptic differential operator and so $X$ is a diffusion process.