Exact packing measure of the range of $ψ$-Super Brownian motions

Abstract

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