Sascha Troscheit: Distinguishing sets and measures from their Brownian images

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, remains separated along a sequence of scales. As an application, we use this to find a criterion for when different self-similar sets (and their natural measures) can be distinguished.