Brownian motion and harmonic functions on rotationally symmetric manifolds. (English) Zbl 0593.60078

March, Peter

Ann. Probab. 14, 793-801 (1986).

Summary: We consider Brownian motion X on a rotationally symmetric manifold \(M_ g=({\mathbb{R}}^ n,ds^ 2),\quad ds^ 2=dr^ 2+g(r)^ 2d\theta^ 2.\) An integral test is presented which gives a necessary and sufficient condition for the nontriviality of the invariant \(\sigma\)-field of X, hence for the existence of nonconstant bounded harmonic functions on \(M_ g\). Conditions on the sectional curvatures are given which imply the convergence or the divergence of the test integral.

MSC:

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

Keywords:

skew product; sectional curvature; rotationally symmetric manifold