PC Seminar: Distinguishing Cantor sets after the Brownian meat grinder

Abstract: Motivated by a question of Mike Hochman, we study whether a single planar Brownian image identifies which of two deterministic time
measures or time sets produced it. We will show that the laws of the Brownian pushforwards of two (reasonable) measures are mutually singular whenever their small scale behaviour, as measured by correlation profiles, remain separated along a sequence of scales. As an application, we use this to find a criterion on when different Cantor sets (and their natural measures) can be distinguished, answering said question by Hochman.

This is a seminar in our seminar series on Probability and Combinatorics (PC).