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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 20th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## 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](http://dx.doi.org/10.1090/s0002-9947-1949-0027960-x). 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](http://dx.doi.org/10.1007/bfb0084322). <a href="https://ncatlab.org/nlab/revision/Feynman-Kac+formula/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Feynman-Kac+formula">current</a>

   Created:

   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.

   v1, current