Regularity and irregularity of superprocesses with $(1+\beta)$-stable branching mechanism

TL;DR

This paper explores the regularity and irregularity properties of densities of super-Brownian motion with $(1+eta)$-stable branching, revealing a dimension-dependent dichotomy and detailed multifractal analysis in one dimension.

Contribution

It provides a comprehensive overview of the regularity properties, including a dichotomy in density behavior across dimensions and multifractal spectrum calculations in one dimension.

Findings

Density is continuous in dimension 1

Density is locally unbounded in higher dimensions

Multifractal spectrum characterized in 1D

Abstract

We would like to give an overview of results on regularity, or better to say "irregularity", properties of densities at fixed times of super-Brownian motion with $(1+\beta)$-stable branching for $\beta<1$. First, the following dichotomy for the density is shown: it is continuous in the dimension $d=1$ and locally unbounded in all higher dimensions where it exists. Then in $d=1$ we determine pointwise and local H\"older exponents of the density, and calculate the multifractal spectrum corresponding to pointwise H\"older exponents.