Diffusion generator ($L=b{\partial }_x+\frac{1}{2}{\sigma }^2{\partial }_{xx}$)

For the one-dimensional Itô equation $dX=b(X)dt+\sigma (X)dW$, the infinitesimal generator on smooth test functions is $L=b{\partial }_x+{\sigma }^2{\partial }_{xx}/2$. The Kolmogorov backward equation is $u_t=Lu$, while its formal adjoint produces the Fokker-Planck equation.

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.