Hit and miss with the density of the (α,β)-superprocess
The (α,β)-superprocess is a spatial branching model associated to
an α-stable spatial motion and a (1+β)-stable branching mechanism.
Technically, 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.