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.


60J65 Brownian motion
58J65 Diffusion processes and stochastic analysis on manifolds
Full Text: DOI