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.
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The Feynman-Kac theorem is a fundamental result in stochastic processes, particularly in the context of linking partial differential equations (PDEs) with stochastic processes, specifically Brownian motion. It provides a way to express the solution of a certain type of PDE in terms of expectations of functionals of stochastic processes, such as those arising from Brownian motion.