Hitting distributions of small geodesic spheres. (English) Zbl 0651.58037

Liao, Ming

Ann. Probab. 16, No. 3, 1039-1050 (1988).

The author generalizes to the n-dimensional case a result of M. A. Pinsky [Ann. Inst. Henri Poincaré, Probab. Stat. 21, 39-46 (1985; Zbl 0559.58026)] on the behaviour of the harmonic measure operator, to geodesic spheres with small radius, for a 2-dimensional Riemannian manifold. In the above cited paper Pinsky’s formula gives as corollary a characterization to surfaces with constant curvature. The analogous result of the author characterizes Einstein manifolds. Also a result is proved which gives us a sufficient condition for scalar curvature to be constant.

Reviewer: J.Ruidival dos Santos Filho

MSC:

58J65	Diffusion processes and stochastic analysis on manifolds
53B20	Local Riemannian geometry
53C20	Global Riemannian geometry, including pinching

Keywords:

Brownian motion; harmonic measure operator; geodesic spheres; scalar curvature

Citations:

Zbl 0559.58026