Brownian motion and the parabolicity of minimal graphs

2008/10/03 by Robert W. Neel, Neel, Robert W. · 1 citation
Economics, Econometrics and Finance · Mathematics · #Mathematical Dynamics and Fractals #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.DG #math.PR #msc:53A10 #msc:58J65 #msc:60H30

paper · pdf · doi:10.48550/arxiv.0810.0669

7 pages

arxiv created 2008/10/03 · arxiv updated 2009/12/01

Abstract

We prove that minimal graphs (other than planes) are parabolic in the sense that any bounded harmonic function is determined by its boundary values. The proof relies on using the coupling introduced in the author's earlier paper "A martingale approach to minimal surfaces" to show that Brownian motion on such a minimal graph almost surely strikes the boundary in finite time.

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• Global maximum principles and divergence theorems on complete manifolds with boundary

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