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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


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.


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Peter March. "Brownian Motion and Harmonic Functions on Rotationally Symmetric Manifolds." Ann. Probab. 14 (3) 793 - 801, July, 1986.


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

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

Primary: 60G65
Secondary: 58G32

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

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 3 • July, 1986
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