Brownian Motion and Harmonic Functions on Rotationally Symmetric Manifolds

Abstract

We consider Brownian motion $X$ on a rotationally symmetric manifold $M_g = (\mathbb{R}^n, ds^2), ds^2 = dr^2 + g(r)^2 d\theta^2$. 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.