# zbMATH — the first resource for mathematics

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