The (α,β)-superprocess is a spatial branching model associated to an α-stable spatial motion and a (1+β)-stable branching mechanism. Formally, it is a measure-valued Markov process, but this talk concerns the absolutely continuous parameter regime, in which the random measure has a density. After introducing this process and some classical results, I will discuss some newly proven path properties of the density. These include (i) strict positivity of the density at a fixed time (for certain values of α and β) and (ii) a classification of the measures which the density “charges” almost surely, and of the measures which the density fails to charge with positive probability, when conditioned on survival. The duality between the superprocess and a fractional PDE is central to our method, and I will discuss how the probabilistic statements above correspond to new results about solutions to the PDE.

**Date**: 14 October 2019, 12:00 (Monday, 1st week, Michaelmas 2019)**Venue**: Mathematical Institute

Woodstock Road OX2 6GGSee location on maps.ox**Details**: L4**Speaker**: Thomas Hughes (UBC)**Organising department**: Department of Statistics**Organiser**: Christina Goldschmidt (Department of Statistics, University of Oxford)**Part of**: Probability seminar**Booking required?**: Not required**Audience**: Public- Editor: Christina Goldschmidt