Open Access
Translator Disclaimer
July, 1986 Brownian Motion and Harmonic Functions on Rotationally Symmetric Manifolds
Peter March
Ann. Probab. 14(3): 793-801 (July, 1986). DOI: 10.1214/aop/1176992438

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.

Citation

Download Citation

Peter March. "Brownian Motion and Harmonic Functions on Rotationally Symmetric Manifolds." Ann. Probab. 14 (3) 793 - 801, July, 1986. https://doi.org/10.1214/aop/1176992438

Information

Published: July, 1986
First available in Project Euclid: 19 April 2007

MathSciNet: MR841584
zbMATH: 0593.60078
Digital Object Identifier: 10.1214/aop/1176992438

Subjects:
Primary: 60G65
Secondary: 58G32

Keywords: invariant $\sigma$-field , sectional curvature , Skew product

Rights: Copyright © 1986 Institute of Mathematical Statistics

JOURNAL ARTICLE
9 PAGES


SHARE
Vol.14 • No. 3 • July, 1986
Back to Top