Global geometry under isotropic Brownian flows. (English) Zbl 1110.60064

Vadlamani, Sreekar; Adler, Robert J.

Electron. Commun. Probab. 11, 182-192 (2006).

Summary: We consider global properties of a codimension one manifold embedded in Euclidean space, as it evolves under an isotropic and volume preserving Brownian flow of diffeomorphisms. In particular, we obtain expressions describing the expected rate of growth of the Lipschitz-Killing curvatures, or intrinsic volumes, of the manifold under the flow. These results shed new light on some of the intriguing growth properties of flows from a global perspective, rather than the local perspective, on which there is a much larger literature.

MSC:

60H20	Stochastic integral equations
52A39	Mixed volumes and related topics in convex geometry
60J65	Brownian motion