Branching diffusions on Hd with variable fission: The hausdorff dimension of the limiting set

M. Kelbert, Iouri M. Soukhov

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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 ∂Hd. Under these assumptions, a formula is established for the Hausdorff dimension h(Λ) of the limiting (random) set Λ ⊆ ∂Hd, 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.

Original languageEnglish (US)
Pages (from-to)155-167
Number of pages13
JournalTheory of Probability and its Applications
Volume51
Issue number1
DOIs
StatePublished - May 1 2007

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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