A note on the empty balls left by a critical branching Wiener process
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A NOTE ON THE EMPTY BALLS LEFT BY A CRITICAL BRANCHING WIENER PROCESS Yueyun Hu (Villetaneuse) Dedicated to Professors Endre Cs´ aki and P´ al R´ev´esz on the occasion of their 70th birthday
Abstract In this note, we partially confirm some conjectures of P. R´ev´esz [10] on the critical branching Wiener process.
1. Introduction The spatial branching process is one of the simplest models that describe a system of particles combining branching property with spatial motion. We consider here a critical branching Wiener process which is denoted by (Zn , n ≥ 0). At time 0, Z0 is a Poisson point measure on Rd whose intensity is the Lebesgue measure: for any measurable A ⊂ Rd , |A|k P # { points of Z0 fall in A } = k = e−|A| , k ≥ 0, k! where |A| denotes the Lebesgue measure of A. Every point of Z0 is associated with a particle which moves, independently of the others, according to the following rules: • a particle starts from x ∈ Rd and executes a d-dimensional Wiener process for a unit time; • arriving at the new location at time 1 the particle dies and gives offsprings: 1 P # offsprings = 0 = P # offsprings = 2 = ; 2 •
each offspring, if exists, starts from where its ancestor died, and executes an independent d-dimensional Wiener process and repeats the above steps, and so on. All Wiener processes and offspring numbers are assumed to be independent; • there is no collision between particles. Mathematics subject classification number: 60J80, 60F15. Key words and phrases: branching Wiener process, empty ball. 0031-5303/2005/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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yueyun hu
Denote by λ(n, x) the number of particles living at time n and at position x ∈ Rd . The process Zn , taking values in positive measures, is defined by λ(n, x)δ{x} . Zn = x
The above sum makes sense because there are only countably many x ∈ Rd such that λ(n, x) > 0. The measure-valued process Zn is called a critical branching Wiener process. The above model and more generally, branching random fields were introduced and studied in detail by R´ev´esz in [9] and in his book [7]. Let us also mention some recent references in various settings: Kesten [5] and R´ev´esz [8] (critical case), Chen [3], R´ev´esz [11] and R´ev´esz, Rosen and Shi [12] (supercritical case), and Cs´aki, R´ev´esz and Shi [4] (coalescing random walk). This note is devoted to the study of the asymptotic behavior of Zn , more precisely the empty balls left by (Zn ). We aim at two conjectures raised in R´ev´esz [10]. Let α > 0 and define def B(x, r) = y ∈ Rd : |y − x| ≤ r , def
B(r) = B(0, r), def R(n) = sup r > 0 : Zn , 1B(r) = 0 , def R(α, n) = sup 0 < r < nα : ∃ x ∈ B(nα − r), Zn , 1B(x,r) = 0 . In other words, R(n) is the radius of the largest ball around the origin which does not contain any particles at time n and R(α, n) is the radius of the largest empty ball contained in B(nα ) at time n. Let us quote the following results in the twodimensional case (R´ev´esz [9], Theorem 6.3 and R´ev´esz [10], T
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