A probabilistic proof of the Gauss-Bonnet-Chern theorem for manifolds with boundary
The authors prove the Gauß-Bonnet-Chern theorem for a manifold with boundary by a probabilistic method. To construct the fundamental solution of the heat equation they adopt a probabilistic approach to the initial value problem due to Airault and Ikeda-Watanabe. Then they combine this result with a modified Malliavin calculus and obtain that the fundamental solution can still be expressed as a generalized Wiener functional expectation. Evaluating probabilistically the short time behaviour of this generalized expectation they can directly compute the Gauß- Bonnet-Chern theorem.
fundamental solution of the heat equation
scientific article; zbMATH DE number 4155531