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Probabilistic methods in differential geometry. (English) Zbl 0698.58051

Stochastic processes, Semin., San Diego/CA (USA) 1989, Prog. Probab. 18, 123-134 (1990).
[For the entire collection see Zbl 0686.00017.]
Let M be a Riemannian manifold with metric g. The metric g determines a second order elliptic operator called the Laplace-Beltrami operator, which locally has the representation \[ \Delta =\frac{1}{\sqrt{\det g}}\frac{\partial}{\partial x^ i}(\sqrt{\det g}g^{ij}\frac{\partial}{\partial x^ j}). \] The process associated with half of the Laplace-Beltrami operator is called the Riemannian Brownian motion on M. The author studies the geometric properties of M by investigating the behavior of the Riemannian Brownian motion, because for some geometric problems the probabilistic approach is nicer and clearer than the analytic approach.
Reviewer: S.Eloshvili

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
53C20 Global Riemannian geometry, including pinching
60J65 Brownian motion
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Citations:

Zbl 0686.00017