# How round are the complementary components of planar Brownian motion?

Monday, January 9, 2017 - 2:30pm
THO 119

Speaker: Nina Holden, Massachusetts Institute of Technology

Abstract: Consider a Brownian motion W in the complex plane started from 0 and run for time 1.
Let A(1), A(2),... denote the bounded connected components of C-W([0,1]).
Let R(i) (resp.\ r(i)) denote the out-radius (resp.\ in-radius) of A(i) for i \in N.
Our main result is that E[\sum_i R(i)^2|\log R(i)|^\theta ]<\infty for any \theta<1.
We also prove that \sum_i r(i)^2|\log r(i)|=\infty almost surely.
These results have the interpretation that most of the components A(i) have a rather regular or round shape.
Based on joint work with Serban Nacu, Yuval Peres, and Thomas S. Salisbury.

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