A stochastic approach to the Poincaré-Hopf theorem (Q1113152)

The Poincaré-Hopf theorem is proved here by a probabilistic method. The proof of this theorem is reduced to an estimate of a supertrace of the fundamental solution. We use the Malliavin calculus to estimate the fundamental solution. Our proof can be applied to the case that the zero point set is the union of the submanifolds under some conditions.