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

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• 作者：Xan Duhalde ; Thomas Duquesne
• 关键词：Super Brownian motion ; General branching mechanism ; Lévy snake ; Total range ; Occupation measure ; Exact packing measure
• 刊名：Probability Theory and Related Fields
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：167
• 期：1-2
• 页码：201-252
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Probability Theory and Stochastic Processes; Theoretical, Mathematical and Computational Physics; Quantitative Finance; Mathematical and Computational Biology; Statistics for Business/Economics/Mathem
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-2064
• 卷排序：167

文摘

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.