## 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 ∂H^{d}. Under these assumptions, a formula is established for the Hausdorff dimension h(Λ) of the limiting (random) set Λ ⊆ ∂H^{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}.

Original language | English (US) |
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Pages (from-to) | 155-167 |

Number of pages | 13 |

Journal | Theory of Probability and its Applications |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - May 1 2007 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty