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Let $X_t$ be a real valued Markov process (starting at $x$) with generator $L$. Let $\Gamma(f)$ denote Carré du champ operator i.e. $L(f^2) - 2f \cdot L (f)$. As far as I know under suitable regularity assumptions: $$ M_t^2 - \int_{0}^{t} \Gamma(f)(X_s) ds $$ is a martingale, where $M_t := f(X_t) - f(x) - \int_{0}^{t}Lf(X_s)ds$ (which is also a martingale by a fairly standard result) . I'm looking for sources which elaborate a little bit on this fact. So far I've only seen proof for diffusion processes. Any help greatly welcomed.

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