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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 $L$?

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  • $\begingroup$ The statement of which you are seeking the converse is not about a formula that holds. Please check this and elaborate what it has to do with the martingale property. $\endgroup$
    – Kurt G.
    Commented Mar 31, 2023 at 15:15
  • $\begingroup$ @KurtG. the statement is theorem 19.21 of link.springer.com/book/10.1007/978-1-4757-4015-8 and is a generalization of the one contained in the link you posted. If it is not true what is the counterexample? $\endgroup$
    – Marco
    Commented Mar 31, 2023 at 16:51

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