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

Description

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

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 210
• Langue: English

Extrait

[DJam]ComptesRendusdel’Acade´miesdesSciences-SeriesIMathe´matiques,Vol332,No12(2001)1053–1058. 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,CorHandn≥2 an integer (n≥3 ifF=R). Let n B n be the Euclidean unit ball ofF. DefineGasF=R:G=SO 0 (n,1),F=C:G= SU(n,1),F=H:G=Sp(n,1); and letG=KANIt isbe its Iwasawa decomposition. well known thatB n can be identified withG/K, in particularGacts onB n andB n can be endowed with aGWe will refer to this metric as the hyperbolic metric on-invariant metric. B n let. Further, d= dim R Fand definem 1 =d(n−1) andm 2 =d−1 the multiplicities of the roots ofG. Ifz∈B n , there existsg∈Gsuch thatg.z= 0 andg.0 =z. Forζ∈B n we will then write ϕ z (ζ) =g.ζso that 2 z−P z ζ−1− kzkQ z ζ ϕ z (ζ) = 1− hζ, zi hζ, zi withP z (ζ) =zandQ z (ζ) =ζ−P z (ζ). hz, zi Denote byD F the Laplace operator onB n that is invariant underG, bydµtheG-invariant 1+α measure onB n . Let Ω be a relatively compact domain inB n of classCsuch that∂Ω =∂Ω. Denote bydσ g the contraction ofdµwith regard to the outward normal vector to∂Ω with respect to the hyperbolic metric. The Green function Γ forD F is then given by Z m 1 1 +m 2 −1 (1−t) 2 Γ(ζ, z) =−c n 1+m+m dt. 2 t kϕ z (ζ)k 2 65