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

Thomas Duquesne 1 Xan Duhalde 2
2 Processus stochastiques
LPMA - Laboratoire de Probabilités et Modèles Aléatoires
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.
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Submitted on : Thursday, July 17, 2014 - 7:00:05 PM
Last modification on : Friday, March 27, 2020 - 3:05:57 AM
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  • HAL Id : hal-01025477, version 1
  • ARXIV : 1407.4913

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Thomas Duquesne, Xan Duhalde. Exact packing measure of the range of $\psi$-Super Brownian motions.. 2014. ⟨hal-01025477⟩

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