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Teor. Veroyatnost. i Primenen., 2006, Volume 51, Issue 1, Pages 241–255 (Mi tvp155)  

This article is cited in 5 scientific papers (total in 5 papers)

Branching diffusions on $H^d$ with variable fission: The Hausdorff dimension of the limiting set

M. Ya. Kelberta, Yu. M. Sukhovbc

a University of Wales Swansea
b Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge
c A. A. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: This paper extends results of previous papers [S. Lalley and T. Sellke, Probab. Theory Related Fields, 108 (1997), pp. 171–192] and [F. I. Karpelevich, E. A. Pechersky, and Yu. M. Suhov, Comm. Math. Phys., 195 (1998), pp. 627–642] on the Hausdorff dimension of the limiting set of a homogeneous hyperbolic branching diffusion to the case of a variable fission mechanism. More precisely, we consider a nonhomogeneous branching diffusion on a Lobachevsky space $H^d$ and assume that parameters of the process uniformly approach their limiting values at the absolute $\partialH^d$. Under these assumptions, a formula is established for the Hausdorff dimension $h(\Lambda)$ of the limiting (random) set $\Lambda\subseteq\partialH^d$, which agrees with formulas obtained in the papers cited above for the homogeneous case. The method is based on properties of the minimal solution to a Sturm–Liouville equation, with a potential taking two values, and elements of the harmonic analysis on $H^d$.

Keywords: Lobachevsky space, branching diffusion, limiting set, Hausdorff dimension, horospheric projection, equidistant projection, Sturm–Liouville equation, minimal positive solution.

DOI: https://doi.org/10.4213/tvp155

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English version:
Theory of Probability and its Applications, 2007, 51:1, 155–167

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Received: 04.09.2005
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Citation: M. Ya. Kelbert, Yu. M. Sukhov, “Branching diffusions on $H^d$ with variable fission: The Hausdorff dimension of the limiting set”, Teor. Veroyatnost. i Primenen., 51:1 (2006), 241–255; Theory Probab. Appl., 51:1 (2007), 155–167

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. Ya. Kelbert, Yu. M. Sukhov, “Large-Time Behavior of a Branching Diffusion on a Hyperbolic Space”, Theory Probab. Appl., 52:4 (2008), 594–613  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Cammarota V., Orsingher E., “Cascades of particles moving at finite velocity in hyperbolic spaces”, J. Stat. Phys., 133:6 (2008), 1137–1159  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Theory Probab. Appl., 57:3 (2013), 419–443  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. Cammarota V. De Gregorio A. Macci C., “On the Asymptotic Behavior of the Hyperbolic Brownian Motion”, J. Stat. Phys., 154:6 (2014), 1550–1568  crossref  mathscinet  zmath  isi  scopus
    5. Orsingher E., Ricciuti C., Sisti F., “Motion Among Random Obstacles on a Hyperbolic Space”, J. Stat. Phys., 162:4 (2016), 869–886  crossref  mathscinet  zmath  adsnasa  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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