Seminar

Analysis & Applications Seminar

Hausdorff Dimension of Caloric Measure

Speaker:  Alyssa Genschaw (Milwaukee School of Engineering)
Date:  Friday, 03 June 2022 - 11:00
Place:  Aula 520, Módulo 17, Departamento de Matemáticas, UAM
Online:  us06web.zoom.us/j/82650507551 (ID: 826 5050 7551)

Abstract:

Caloric measure is a probability measure supported on the boundary of a domain in \(R^{n+1}=R^n\times R\) (space \(\times\) time) that is related to the Dirichlet problem for the heat equation in a fundamental way. Equipped with the parabolic distance, \(R^{n+1}\) has Hausdorff dimension \(n+2\). We prove that (even on domains with geometrically very large boundary), the caloric measure is carried by a set of Hausdorff dimension at most \(n+2-\beta_n\) for some \(\beta_n>0\). The corresponding theorem for harmonic measure is due to Bourgain (1987), but the proof in that paper contains a gap.

Additionally, we prove a caloric analogue of Bourgain's alternative. I will briefly discuss the results, including how we fix the gap in the original proof. This is joint work with Matthew Badger.