Homotopy and Homology Vanishing Theorems and the Stability of Stochastic Flows

Details Homotopy and Homology Vanishing Theorems and the Stability of Stochastic Flows Homotopy and Homology Vanishing Theorems and the Stability of Stochastic Flows

1995-03-22

arxiv.org/abs/dg-ga/9503008v1

Mathematics

Differential Geometry

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Scientific paper

We relate stability properties (i.e. moment exponents) of a stochastic dynamical system on a compact manifold $M$ to the homotopy and integral homology groups of $M$. In the special case of gradient Brownian systems associated to isometric immersions of $M$ in Euclidean space, these moment exponents can be estimated in terms of the second fundamental form of the immersion. This yields topological obstructions to isometric immersions generalizing results of Lawson-Simons and others. Our work also places these authors' work into the general framework of Weitzenb\"ock formulas.

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