Homotopy and Homology Vanishing Theorems and the Stability of Stochastic Flows

TL;DR

This paper connects the stability of stochastic flows on manifolds to topological invariants like homotopy and homology groups, providing new obstructions to isometric immersions and linking to classical geometric formulas.

Contribution

It establishes a novel relationship between stochastic stability properties and topological invariants, extending classical results and framing them within Weitzenböck formulas.

Findings

Topological invariants influence stochastic stability exponents.

Obstructions to isometric immersions are derived from homotopy and homology.

Framework unifies stochastic stability with classical differential geometry.

Abstract

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.