Occupation time fluctuations of an infinite variance branching system in large dimensions

a r X i v :m a t h /0511745v 1 [m a t h .P R ] 30 N o v 2005OCCUPATION TIME FLUCTUATIONS OF AN INFINITE VARIANCE

BRANCHING SYSTEM IN LARGE DIMENSIONS

(OCCUPATION TIME FLUCTUATIONS IN LARGE DIMENSIONS)TOMASZ BOJDECKI Institute of Mathematics University of Warsaw ul.Banacha 202-097Warsaw,Poland E.mail:tobojd@mimuw.edu.pl LUIS G.GOROSTIZA ∗Department of Mathematics Centro de Investigaci´o n y de Estudios Avanzados A.P.14-740M´e xico,07000D.F.,Mexico E.mai:lgorosti@math.cinvestav.mx ANNA TALARCZYK Institute of Mathematics University of Warsaw ul.Banacha 202-097Warsaw,Poland E.mail:annatal@mimuw.edu.pl We prove limit theorems for rescaled occupation time fluctuations of a (d,α,β)-branching particle system (particles moving in R d according to a spherically symmetric α-stable L´e vy process,(1+β)-branching,0<β<1,uniform Poisson initial state),in the cases of critical dimension,d =α(1+β)/β,

and large dimensions,d >α(1+β)/β.The fluctuation processes are continuous but their limits are stable processes with independent increments,which have jumps.The convergence is in the sense of finite-dimensional distributions,and also of space-time random fields (tightness does not hold in the usual Skorohod topology).The results are in sharp contrast with those for intermediate dimensions,α/β

Key words:branching particle system,critical and large dimensions,limit theorem,occupation time fluctuation,stable process.

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