Summary
In this paper, the Poincaré-Hopf theorem (Hopf [8]) is proved by a probabilistic method. The proof of this theorem is reduced to an estimate of a supertrace of the fundamental solution. We use the Malliavian calculus to estimate the fundametal solution. Our proof can be applied to the case that the zero point set is the union of the submanifolds under some conditions.
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Ueki, N. A Stochastic approach to the Poincaré-Hopf theorem. Probab. Th. Rel. Fields 82, 271–293 (1989). https://doi.org/10.1007/BF00354764
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DOI: https://doi.org/10.1007/BF00354764
Keywords
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Fundamental Solution
- Probabilistic Method