Exact packing measure of the range of $\psi$-Super Brownian motions

TL;DR

This paper determines the exact packing measure of the range of $ ext{psi}$-super Brownian motions in high dimensions, generalizing previous quadratic branching results and linking it to the process's branching mechanism.

Contribution

It provides a precise formula for the packing measure of the process's range for a broad class of branching mechanisms, extending earlier quadratic case findings.

Findings

The total range has an exact packing measure with a specific gauge function.

The occupation measure equals the packing measure up to a constant.

The packing dimension matches the critical threshold for the process.

Abstract

We consider super processes whose spatial motion is the $d$-dimensional Brownian motion and whose branching mechanism $\psi$ is critical or subcritical; such processes are called $\psi$-super Brownian motions. If $d\!>\!2\bgamma/(\bgamma\!-\!1)$, where $\bgamma\!\in\!(1,2]$ is the lower index of $\psi$ at $\infty$, then the total range of the $\psi$-super Brownian motion has an exact packing measure whose gauge function is $g(r)\! =\! (\log\log1/r) / \varphi^{-1} ( (1/r\log\log 1/r)^{2})$, where $\varphi\! =\! \psi^\prime\! \circ \! \psi^{\!-1}$. More precisely, we show that the occupation measure of the $\psi$-super Brownian motion is the $g$-packing measure restricted to its total range, up to a deterministic multiplicative constant only depending on $d$ and $\psi$. This generalizes the main result of \cite{Duq09} that treats the quadratic branching case. For a wide class of $\psi$, the constant $2\bgamma/(\bgamma\!-\!1)$ is shown to be equal to the packing dimension of the total range.