Dynkin formula for a diffusion (${\mathbb{E}}_x\phi (X_T)=\phi (x)+{\mathbb{E}}_x{\int }_0^{T}L\phi (X_s)ds$)

For a diffusion process with diffusion generator $L$, the Itô formula makes $\phi (X_t)-\phi (X_0)-{\int }_0^{t}L\phi (X_s)ds$ a local martingale. Under stopping and integrability conditions making it a true martingale, taking expectations gives the displayed identity. For Brownian motion, $L=\frac{1}{2}\Delta$, recovering the usual Brownian formula.