Singularities of stable super-Brownian motion

TL;DR

This paper studies the spectrum of singularities in super-Brownian motion with stable branching, providing a comprehensive description in high dimensions and partial insights in lower dimensions, enhancing understanding of its complex behavior.

Contribution

It offers a uniform characterization of singularities across dimensions and identifies times with higher-order singularities, extending previous research on density.

Findings

Singularities are characterized at every time in high dimensions.

Higher-order singularities occur at specific random times.

Partial descriptions of singularities in lower dimensions are provided.

Abstract

We investigate in this work the spectrum of singularities of super-Brownian motion with stable branching. The main purpose is to provide a uniform description of the latter in high dimension $d\geq\tfrac{2}{\gamma-1}$, presenting the singularities existing at every time t and characterising as well the set of random times at which singularities of higher order appear. In lower dimensions, we give a partial description of the singularities which complement the recent study of the density by Mytnik and Wachtel.