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Brownian motion, random walks on trees, and harmonic measure on polynomial Julia sets

Nathaniel D. Emerson
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Abstract

We consider the harmonic measure on a disconnected polynomial Julia set in terms of Brownian motion. We show that the harmonic measure of any connected component of such a Julia set is zero. Associated to the polynomial is a combinatorial model, the tree with dynamics. We define a measure on the tree, which is a combinatorial version on harmonic measure. We show that this measure is isomorphic to the harmonic measure on the Julia set. The measure induces a random walk on the tree, which is isomorphic to Brownian motion in the plane.

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Nathaniel D. Emerson. 2006-09-12. Brownian motion, random walks on trees, and harmonic measure on polynomial Julia sets. https://arxiv.org/abs/math/0609044

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