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

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 $d\text{\hspace{0.17em}⁣}>\text{\hspace{0.17em}⁣}2\text{bgamma}\mathrm{/}(\text{bgamma}\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}1)$, where $\text{bgamma}\text{\hspace{0.17em}⁣}\in \text{\hspace{0.17em}⁣}(1,2]$ is the lower index of $\psi$ at $\mathrm{\infty }$, then the total range of the $\psi$-super Brownian motion has an exact packing measure whose gauge function is $g®! =! (\log\log1/r) / \varphi^{-1} ( (1/r\log\log 1/r)^{2})$,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 $2\text{bgamma}\mathrm{/}(\text{bgamma}\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}1)$ is shown to be equal to the packing dimension of the total range.