How round are the complementary components of planar Brownian motion?

Nina Holden, Serban Nacu, Yuval Peres, Thomas S. Salisbury

Research output: Contribution to journalArticlepeer-review


Consider a Brownian motion W in C 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 ? N. Our main result is that E[ i R(i)2| log R(i)|?] < ∞ for any Θ < 1. We also prove that Σi r(i)2| log r(i)| = ∞ almost surely. These results have the interpretation that most of the components A(i) have a rather regular or round shape.

Original languageEnglish (US)
Pages (from-to)882-908
Number of pages27
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number2
StatePublished - May 1 2019


  • Complementary components of planar Brownian motion
  • Planar Brownian motion

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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