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

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 $g(r)=(\log \log 1/r)/{\phi }^{-1}((1/r\log \log 1/r{)}^2)$, where $\phi ={\psi }^{\prime }\circ {\psi }^{-1}$. 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.

43 pages