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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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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
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Cammarota V., Orsingher E., “Cascades of particles moving at finite velocity in hyperbolic spaces”, J. Stat. Phys., 133:6 (2008), 1137–1159
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Theory Probab. Appl., 57:3 (2013), 419–443
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Cammarota V. De Gregorio A. Macci C., “On the Asymptotic Behavior of the Hyperbolic Brownian Motion”, J. Stat. Phys., 154:6 (2014), 1550–1568
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Orsingher E., Ricciuti C., Sisti F., “Motion Among Random Obstacles on a Hyperbolic Space”, J. Stat. Phys., 162:4 (2016), 869–886
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