### Abstract

Consider a time-and space-homogeneous random branching Markov process on a d-D Lobachevsky space H_{d}. Its asymptotic behaviour can be described in terms of the Hausdorff dimension of the (random) set Λ of the accumulation points (on the absolute ∂H_{d}). The simplest and most well-known example is the Laplace-Beltrami branching diffusion; in the case d = 2 the Hausdorff dimension of Λ was calculated in [LS]. In this paper we extend the formula for the Hausdorff dimension to d ≥ 3 and a larger class of branching processes. It turns out that the Hausdorff dimension of Λ takes either a value from (O, (d - 1)/2) or equals d - 1, the Euclidean dimension of ∂H_{d}, which gives an interesting exmaple of a "geometric" phase transition.

Original language | English (US) |
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Pages (from-to) | 627-642 |

Number of pages | 16 |

Journal | Communications In Mathematical Physics |

Volume | 195 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 1998 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications In Mathematical Physics*,

*195*(3), 627-642. https://doi.org/10.1007/s002200050405