Reflecting Brownian Motion and the Gauss–Bonnet–Chern Theorem

Weitao Du , Elton P. Hsu

Communications in Mathematics and Statistics ›› 2023, Vol. 11 ›› Issue (3) : 609 -627.

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Communications in Mathematics and Statistics ›› 2023, Vol. 11 ›› Issue (3) : 609 -627. DOI: 10.1007/s40304-021-00266-3
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Reflecting Brownian Motion and the Gauss–Bonnet–Chern Theorem

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Abstract

We use reflecting Brownian motion (RBM) to prove the well-known Gauss–Bonnet–Chern theorem for a compact Riemannian manifold with boundary. The boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.

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Manifold with boundary / Gauss–Bonnet–Chern theorem / Reflecting Brownian motion

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Weitao Du, Elton P. Hsu. Reflecting Brownian Motion and the Gauss–Bonnet–Chern Theorem. Communications in Mathematics and Statistics, 2023, 11(3): 609-627 DOI:10.1007/s40304-021-00266-3

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