## STABLE LAWS ARISING FROM HITTING DISTRIBUTIONS OF PROCESSES ON HOMOGENEOUS TREES AND THE HYPERBOLIC HALF-PLANE (2001)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Baldi01stablelaws,

author = {Paolo Baldi and Enrico Casadio Tarabusi and Alessandro Figà-talamanca},

title = {STABLE LAWS ARISING FROM HITTING DISTRIBUTIONS OF PROCESSES ON HOMOGENEOUS TREES AND THE HYPERBOLIC HALF-PLANE},

year = {2001}

}

### OpenURL

### Abstract

The projective line with respect to a local field is the boundary of the Bruhat-Tits tree associated to the field, much in the same way as the realprojective line is the boundary of the upper half-plane. In both cases we may consider the horocycles with respect to the point at infinity. These horocycles are exactly the horizontal lines {y = a} with a>0 in the real case, while in the case of a local field the horocycles may be thought of as discretizations of the field obtained by collapsing to a point each ball of a given radius. In this paper we exploit this geometric parallelism to construct symmetric α-stable random variables on the real line and on a local field by completely analogous procedures. In the case of a local field the main ingredient is a drifted random walk on the tree. In the real case the random walk is replaced by a drifted Brownian motion on the hyperbolic halfplane.

### Citations

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Citation Context ...the drift coefficient. 1. Introduction. There are many equivalent definitions of a stable symmetric random variable with values in the real field (for detailed treatments see [GK], or the more recent =-=[ST]-=-). That which most easily extends to local fields is based on the form of its characteristic function: 257258 P. BALDI, E. CASADIO TARABUSI AND A. FIG À-TALAMANCA Definition 1.1. A symmetric random v... |

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Citation Context ...y an explicit formula to the drift coefficient. 1. Introduction. There are many equivalent definitions of a stable symmetric random variable with values in the real field (for detailed treatments see =-=[GK]-=-, or the more recent [ST]). That which most easily extends to local fields is based on the form of its characteristic function: 257258 P. BALDI, E. CASADIO TARABUSI AND A. FIG À-TALAMANCA Definition ... |

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Citation Context ... infinite variance, but is in the extended attraction domain of a Gaussian law. These results are closely linked with results on exponential functionals of Brownian motions first obtained by Dufresne =-=[D]-=-, motivated by applications to risk theory. We give a short alternate proof of Dufresne’s result in the context of the hyperbolic half-plane. The authors are aware of the fact that the results on rand... |

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Citation Context ... 0 e (2) 2W s −2νs ds, Y 2 ) y,s ds = γ(y 2 A ν t ), where γ(t) is a Brownian motion independent of W (1) t ,W (2) t .ThusXy,∞ has the same law as γ(y2Aν ∞). By a result of Dufresne [D] (see also Yor =-=[Y]-=-), the random variable Aν ∞ has the same law as the reciprocal of 2Z, where Z is a random variable having law Γ(ν, 1). That is, Z has a density ⎧ ⎨z g(z) = ⎩ ν−1e−z for z>0, (4.3) Γ(ν) 0for z ≤ 0. The... |

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Citation Context ...α-stable symmetric random variable Ut such that E[χξ(Ut)] = E[χξ(U)] t , as if Ut were obtained multiplying U by t1/α . Thus we can define an α-stable process on F with respect to real time (see also =-=[V]-=-). The relationship between the parameter α and the random walk used to derive U should also clarify the role of the geometry of the field and its Bruhat-Tits tree in the definition of α-stability. In... |

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Citation Context ...fine an α-stable symmetric random variable to be a random variable U with values in F whose characteristic function is of the type E[χξ(U)] = e −c|ξ|α (1.2) for all ξ ∈F, for some c>0and some α>0(cf. =-=[I]-=-, [K]). This formal definition does not give any clue on the possible role of stable random variables in the definition of diffusion processes, neither does it immediately account for the role of the ... |

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Citation Context ...ounded random variable with characteristic function identically 1 on a compact open subgroup of F. In some respect this should be considered the analogue of a Gaussian random variable in this context =-=[E]-=-. In the second part of the paper we study a similar continuous-time situation. The role of the Bruhat-Tits tree of a local field is now played by the hyperbolic upper half-plane, and that of the rand... |

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Citation Context ...an α-stable symmetric random variable to be a random variable U with values in F whose characteristic function is of the type E[χξ(U)] = e −c|ξ|α (1.2) for all ξ ∈F, for some c>0and some α>0(cf. [I], =-=[K]-=-). This formal definition does not give any clue on the possible role of stable random variables in the definition of diffusion processes, neither does it immediately account for the role of the “geom... |

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Citation Context ...e larger is α, the more probably we move into a smaller ball rather than out to a larger one. (The vertices of the BruhatTits tree are the balls of F, see §2.) This intuitive content is elaborated in =-=[F1]-=- in the context of finite ultrametric spaces, with compact ultrametric spaces as the limiting case. It also turns out that the limitation α ≤ 2 is no longer in effect on a local field. This may actual... |

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