## The Annals of Probability

### Brownian Motion and Harmonic Functions on Rotationally Symmetric Manifolds

Peter March

#### 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.

#### Article information

Source
Ann. Probab., Volume 14, Number 3 (1986), 793-801.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176992438

Digital Object Identifier
doi:10.1214/aop/1176992438

Mathematical Reviews number (MathSciNet)
MR841584

Zentralblatt MATH identifier
0593.60078

JSTOR