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Probability Theory and Related Fields

, Volume 167, Issue 1–2, pp 201–252 | Cite as

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

  • Xan Duhalde
  • Thomas DuquesneEmail author
Article
  • 122 Downloads

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.

Keywords

Super Brownian motion General branching mechanism Lévy snake Total range Occupation measure Exact packing measure 

Mathematics Subject Classification

Primary 60G57 60J80 Secondary 28A78 

Notes

Acknowledgments

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

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Sorbonne Universités, UPMC Université Paris 06, LPMA (UMR 7599)Paris Cedex 05France

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