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Recovery of a manifold by a Brownian traveller. (English) Zbl 0571.60087

Let \(X_ t(\omega)\) denote the paths of a process with infinitesimal generator \(L=2^{-1}\Delta +V\) on a compact Riemannian \(C^{\infty}\)-manifold M (which should be assumed to be connected), \(\Delta\) the Laplace-Beltrami operator associated with the Riemannian metric: ”Brownian motion with drift”.
Theorem 1: Knowing the topological space M and one path \(X.(\omega):{\mathbb{R}}^+\to M\) \((\omega \in \Omega_ 1\) with full measure), the differentiable structure, the Riemannian metric and the vector field V can be reconstructed.
Theorem 2: For \(\omega\) as in theorem 1, \[ C_{\omega}:=\left\{\begin{matrix} {\mathbb{R}}^+\to {\mathbb{R}}\\ s\to f(X_ s(\omega))\end{matrix} \mid \quad f\in C(M)\right\} \] determines the topological space M, the differentiable structure, the Riemannian metric and the vector field V up to isomorphism.
Reviewer: H.Crauel

MSC:

60J65 Brownian motion
58J65 Diffusion processes and stochastic analysis on manifolds
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