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Thomas S. Salisbury - The complement of the planar Brownian path

THOMAS S. SALISBURY, Department of Mathematics and Statistics, York University, Toronto, Ontario  M3J 1P3, Canada
The complement of the planar Brownian path

Take a planar Brownian path, run until it exits the unit ball. The complement of the path consists of many components, and we can ask about their general shape. The talk will describe joint work with Yuval Peres, in which we show that the components are round, in the sense that their areas are comparable to the square of their diameters. More formally, we show that for every $\beta>0$, $\sum A_i^\beta<\infty$ if and only if $\sum R_i^{2\beta}<\infty$, where Ai is the area of the i-th component, and Ri is its diameter.