The square of the field is a quadratic variation

Question

I'm looking for sources which elaborate a little bit on the fact that for Markov process $X_t$ with generator $L$, $\int_{}^{}\Gamma(f,f)(X_s)ds$ is a quadratic variation of $M_t := f(X_t) - f(x) - \int_{}^{t}Lf(X_s)ds$ (i.e. $M_t^2 - \int_{}^{}\Gamma(f,f)(X_s)ds$ is a martingale) where $\Gamma(f,f) = Lf^2 - 2fL(f)$. Thank you for all suggestions.

Answer 1

Could these papers be of interest ?

http://djalil.chafai.net/blog/2017/04/09/carre-du-champ/

Show that the carré du champ operator is nonnegative

https://mathoverflow.net/questions/229226/carre-du-champ-subunit-paths-and-cc-metrics

http://www.numdam.org/article/SB_1976-1977__19__167_0.pdf

Answer 2

Check out Proposition 3.26 in https://andre-schlichting.de/wp-content/uploads/2012/09/MP-script.pdf