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.