It is a well known (see of theorem 19.21 of this book) that any Feller stochastic process $X$ with infinitesimal generator $(\mathcal{L},\mathcal{D})$ satisfies the Dynkin's formula i.e. for any $f \in \mathcal{D}$ $$ M^f_t:=f(X_t)-f(X_0)-\int_0^t \mathcal{L}f(X_s) ds $$ is a martingale.
Is it true also the converse? Specifically, If $L$ is an elliptic (or hypoelliptic) second order operator and $X_t$ is a Markov process for which the formula above holds is it true that $X_t$ is a Feller process with infinitesimal generator $L$?