Dynkin formula for a diffusion
= Dynkin formula for a diffusion
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{title2=$\mathbb E_x\varphi(X_T)=\varphi(x)+\mathbb E_x\int_0^TL\varphi(X_s)ds$}
For a <diffusion process> with <diffusion generator> $L$, the <Itô formula> makes $\varphi(X_t)-\varphi(X_0)-\int_0^tL\varphi(X_s)ds$ a local martingale. Under stopping and integrability conditions making it a true martingale, taking expectations gives the displayed identity. For <Brownian motion>, $L=\tfrac12\Delta$, recovering the usual Brownian formula.