nLab Feynman-Kac formula

Idea

The Feynman–Kac formula expresses the integral kernel of the one-parameter semigroup generated by a Laplacian on a smooth manifold as the path integral of the parallel transport map associated to the given connection with respect to all paths of a given length connecting the two given points.

References

The original reference is

• Mark Kac, On distributions of certain Wiener functionals, Transactions of the American Mathematical Society 65:1 (1949), 1–13. doi.

The case of smooth manifolds is treated in

• James R. Norris, A complete differential formalism for stochastic calculus in manifolds, Séminaire de Probabilités XXVI, Lecture Notes in Mathematics (1992), 189–209. doi.