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Homotopy and homology vanishing theorems and the stability of stochastic flows. (English) Zbl 0858.58053

Let \(M\) be a compact smooth manifold. Let \((F_t)_{t \geq 0}\) be the flow of a stochastic differential equation (SDE) on \(M\). For \(p\in \mathbb{R}\) and \(q\in \mathbb{N}\) put \[ \mu^q (p)=\limsup_{t\to\infty} {1\over t} \log\sup_{x \in M} E\bigl(|\wedge^qT_xF_t |^p \bigr) \bigl(\text{so }\mu^q(p) \leq \mu^1(pq) \text{ for } p \geq 1\bigr); \] here \(|\cdot |\) is induced by an arbitrary choice of a Riemann metric on \(M\). It is shown that \(\pi_1 (M)=0\), \(\pi_2 (M)=0\), and \(H_q(M; \mathbb{Z})=0\), resp., if there exists an SDE on \(M\), whose flow \(F_t\) satisfies \(\mu^1(1)<0\), \(\mu^2(1)<0\), and \(\mu^q(1)<0\), respectively. In particular, \(\mu ([{\dim M+1 \over 2}]) < 0\) implies that \(M\) is a homotopy sphere. Then the special case of gradient Brownian systems, given by isometric immersions of \(M\) into \(\mathbb{R}^m\), is considered. It is shown that positivity of the Schrödinger operator with a potential \(h^q_p\), which is determined by the second fundamental form of the immersion, implies \(\mu^q(p)<0\) for the gradient Brownian system.

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
57N65 Algebraic topology of manifolds
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References:

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