Hausdorff Dimension of Caloric Measure

  • Thu 3 Mar 22

    15:00 - 16:00

  • Online


  • Event speaker

    Alyssa Genschaw

  • Event type

    Lectures, talks and seminars

  • Event organiser

    Mathematics, Statistics and Actuarial Science, School of

  • Contact details

    Dmitry Savostyanov

These Departmental Seminars are for everyone in Maths. We encourage anyone interested in the subject in general, or in the particular subject of the seminar, to come along. It's a great opportunity to meet people in the Maths Department and join in with our community.

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.


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)