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

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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 \(d>2\varvec{\gamma }/(\varvec{\gamma }-1)\), where \(\varvec{\gamma }\in (1,2]\) 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 \(g(r) = (\log \log 1/r) / \varphi ^{-1} ( (1/r\log \log 1/r)^{2})\), 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 Duquesne (Ann Probab 37(6):2431–2458, 2009) that treats the quadratic branching case. For a wide class of \(\psi \), the constant \(2\varvec{\gamma }/(\varvec{\gamma }-1)\) is shown to be equal to the packing dimension of the total range.

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Acknowledgments

We warmly thank J.-F. Delmas, E. Perkins and an anonymous referee for several comments that improved a first version of this article.

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  1. Sorbonne Universités, UPMC Université Paris 06, LPMA (UMR 7599), Boîte courrier 188, 4 place Jussieu, 75252, Paris Cedex 05, France

    Xan Duhalde & Thomas Duquesne

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  1. Xan Duhalde
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  2. Thomas Duquesne
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Correspondence to Thomas Duquesne.

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Duhalde, X., Duquesne, T. Exact packing measure of the range of \(\psi \)-Super Brownian motions. Probab. Theory Relat. Fields 167, 201–252 (2017). https://doi.org/10.1007/s00440-015-0680-2

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  • DOI: https://doi.org/10.1007/s00440-015-0680-2

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