Kolmogorov backward equation

For a diffusion with infinitesimal generator $L$, the Kolmogorov backward equation is ${\partial }_{t}u=Lu$. Its solutions propagate terminal observables backward through the transition semigroup.

The Feynman-Kac formula represents solutions of certain parabolic partial differential equations as conditional expectations of functionals of a diffusion. It follows by applying Itô formula to the solution along the diffusion and taking expectations.