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The number of generations entirely visited for recurrent random walks on random environment

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8 Pages
English

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Niveau: Supérieur, Doctorat, Bac+8
The number of generations entirely visited for recurrent random walks on random environment P. Andreoletti, P. Debs ? December 16, 2011 Abstract In this paper we are interested in a random walk in a random environment on a super-critical Galton-Watson tree. We focus on the recurrent cases already studied by Y. Hu and Z. Shi [7], [6], G. Faraud, Y. Hu and Z. Shi [5], and G. Faraud [4]. We prove that the largest generation entirely visited by these walks behaves like logn and that the constant of normalization which differs from a case to another is function of the inverse of the constant of Biggins' law of large number for branching random walks [1]. 1 Introduction and results First, let us define the process: The environment E: Let T0 a N0-ary regular tree rooted at ?. For all vertices x ? T0 we associate a random vector (A(x1), A(x2), · · · , A(xNx), Nx) where Nx is a non-negative integer bounded by N0. We assume that the sequence (A(x1), A(x2), · · · , A(xNx), Nx), x ? T0) is i.i.

  • galton watson tree

  • largest generation

  • tree rooted

  • super-critical galton-watson tree

  • following constant

  • entirely visited


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[DJam]ComptesRendusdelAcade´miesdesSciences-SeriesIMathe´matiques,Vol332,No12(2001)10531058. EstimationsdunoyaudeGreen,proprie´te´devaleurmoyenne etg´eom´etriedesbouleshyperboliques
Cyrille DOMENICHINO & Philippe JAMING
LAPT(UMR6632),Universit´edeProvence,39,rueJoliot-Curie,13453Marseillecedex13, FRANCE
R´esume´:´mitseseayonudsesoou,ntesdonenbtDnacsteetonreenudeG danslesbouleshyperboliquesr´eelles,complexesetquaternioniques.Celles-ci nous permettent ensuite de montrer que, dans ces boules, les seuls domaines 1+α de classeC,α >cifauresvrsteequeiaesq0upourl´egelsle´edlatieynnalom pourtouteslesfonctionsharmoniquessontlesboulesg´eod´esiques. English title :Green kernel estimates, mean value properties and geometry of classical rank one balls. English abstract :In this Note, we obtain estimates of the Green kernel of real, complex and quaternionic hyperbolic balls. We then apply these to show 1+α that in such balls the only domains of classC,α >0 for which the spherical mean value identity holds for every harmonic function are the geodesic balls.
English Abridged Version In this Note, we denote byF=R,CorHandn2 an integer (n3 ifF=R). Let n Bnbe the Euclidean unit ball ofF. DefineGasF=R:G=SO0(n,1),F=C:G= SU(n,1),F=H:G=Sp(n,1); and letG=KANIt isbe its Iwasawa decomposition. well known thatBncan be identified withG/K, in particularGacts onBnandBncan be endowed with aGWe will refer to this metric as the hyperbolic metric on-invariant metric. Bnlet. Further, d= dimRFand definem1=d(n1) andm2=d1 the multiplicities of the roots ofG. IfzBn, there existsgGsuch thatg.z= 0 andg.0 =z. ForζBnwe will then write ϕz(ζ) =g.ζso that 2 zPzζ1− kzkQzζ ϕz(ζ) = 1− hζ, zi hζ, zi withPz(ζ) =zandQz(ζ) =ζPz(ζ). hz, zi Denote byDFthe Laplace operator onBnthat is invariant underG, bytheG-invariant 1+α measure onBn. Let Ω be a relatively compact domain inBnof classCsuch thatΩ =Ω. Denote bygthe contraction ofwith regard to the outward normal vector toΩ with respect to the hyperbolic metric. The Green function Γ forDFis then given by Z m 1 1 +m21 (1t) 2 Γ(ζ, z) =cn1+m+mdt. 2 t kϕz(ζ)k2 65