Hausdorff Dimension of Caloric Measure

Hausdorff Dimension of Caloric Measure

Caloric measure is a probability measure supported on the boundary of a domain in R^{n+1} = R^n × R (space × 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.

Speaker

Alyssa Genschaw, Milwaukee School of Engineering

How to attend

If not a member of the Dept. Mathematical Science at the University of Essex, you can register your interest in attending the seminar and request the Zoom’s meeting password by emailing Dr Dmitry Savostyanov (d.savostyanov@essex.ac.uk)