Optimal local H\"{o}lder index for density states of superprocesses with $(1+\beta)$-branching mechanism

TL;DR

This paper investigates the regularity of density functions of super-stable processes with branching, establishing conditions for local Hölder continuity and identifying the optimal Hölder index in certain cases.

Contribution

It provides a precise characterization of when the density of super-stable processes is Hölder continuous and determines the optimal Hölder index for these densities.

Findings

Density is Hölder continuous if $d=1$ and $ ext{α} > 1+eta$

Density is unbounded otherwise

Optimal Hölder index is explicitly determined in the continuous case

Abstract

For $0<\alpha\leq2$, a super-$\alpha$-stable motion $X$ in $\mathsf{R}^d$ with branching of index $1+\beta\in(1,2)$ is considered. Fix arbitrary $t>0$. If $d<\alpha/\beta$, a dichotomy for the density function of the measure $X_t$ holds: the density function is locally H\"{o}lder continuous if $d=1$ and $\alpha>1+\beta$ but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local H\"{o}lder index.