Auxiliary facts related to the stable density Scheme of proofs

Particle systems with quasi-homogeneous initial states and their occupation time fluctuations 197 3 Proofs

3.1 Auxiliary facts related to the stable density

We recall the self-similarity property of p t , p at x = a −1α p t a −1α x, x ∈ R, a 0. 3.1 and the bound p 1 x ≤ C 1 + |x| 1+ α . 3.2 Since p t · is decreasing on R + and symmetric, then by 3.1 we have p t x + y ≤ g t x := ¨ t −1α p 1 0, if |x| ≤ 2 p t x 2 , if |x| 2 , x ∈ R, | y| ≤ 1. 3.3 Denote φ m x = 1 1 + |x| m , m 0. 3.4 For ϕ ∈ S R, |ϕx| ≤ Cϕ, mφ m x. This, and an obvious inequality, 1 1 + |x + y| m ≤ Cm 1 + | y| m 1 + |x| m , m 0, 3.5 imply that for any n ∈ N, t 1 , . . . , t n 0, ϕ 1 , . . . , ϕ n ∈ S R and non-negative, we have T t 1 ϕ 1 T t 2 ϕ 2 . . . T t n ϕ n . . . x + y ≤ CT t 1 φ m T t 2 φ m . . . T t n φ m . . . x, | y| ≤ a, m 0, 3.6 where the constant C depends on m, a, ϕ 1 , . . . , ϕ n .

3.2 Scheme of proofs

As announced in the Introduction, we will give a detailed proof of Theorem 2.2a only, neverthe- less, it seems worthwhile to present a general scheme, based on the central limit theorem, which can be applied to the proofs of all parts of Theorems 2.2 and 2.4 see [7] for details. Let N x denote the empirical process of the system with or without branching started from a single particle at x, and let N j , j ∈ Z, be the empirical process for the particles which at time t = 0 belong to [ j, j + 1, i.e., N j = θ j X n=1 N κ j,n , 3.7 according to the description at the beginning of Section 2 see 2.2. Note that N j , j ∈ Z, are independent. The process X T defined in 1.1 can be written as X T t = X j ∈Z 1 F T Z T t N j s − EN j s ds. 3.8 The first step in the argument is to prove that for any ϕ, ψ ∈ S R, and s, t ≥ 0, lim T →∞ E 〈X T t, ϕ〉〈X T s, ψ〉 = E〈X t, ϕ〉〈X s, ψ〉, 3.9 198 Electronic Communications in Probability where X is the corresponding limit process. Without loss of generality we may assume that ϕ, ψ ≥ 0. Using 3.8 we have E 〈X T t, ϕ〉〈X T s, ψ〉 = X j ∈Z 1 F 2 T Z T t Z Ts E 〈N j r , ϕ〉〈N j r ′ , ψ〉d r ′ d r − X j ∈Z 1 F 2 T Z T t Z Ts E 〈N j r , ϕ〉E〈N j r ′ , ψ〉d r ′ d r. 3.10 Using 3.7, 2.1 and the fact that E 〈N x t , ϕ〉 = T t ϕx in both non-branching and critical branching cases, and defining, for x ∈ R, n ≤ k, random variables h k,n x = ρ [x] k,n − x, 3.11 where [x] is the largest integer ≤ x, we rewrite 3.10 as E 〈X T t, ϕ〉〈X T s, ψ〉 = ∞ X k=0 p k k X n=1 I T ; k, n + ∞ X k=0 p k k X n,m=1 n 6=m II T ; k, n, m − ∞ X k=0 p k k X n=1 ∞ X ℓ=0 p ℓ ℓ X m=1 III T ; k, n; ℓ, m, 3.12 where I T ; k, n = 1 F 2 T Z T t Z Ts X j ∈Z E 〈N ρ j k,n r , ϕ〉〈N ρ j k,n r ′ , ψ〉d r ′ d r = 1 F 2 T Z T t Z Ts Z R E 〈N x+h k,n x r , ϕ〉〈N x+h k,n x r ′ , ψ〉d x d r ′ d r, 3.13 II T ; k, n, m = 1 F 2 T Z T t Z Ts X j ∈Z E T r ϕρ j k,n T r ′ ψρ j k,m d r ′ d r = 1 F 2 T Z T t Z Ts Z R E € T r ϕx + h k,n xT r ′ ψx + h k,m x Š d x d r ′ d r 3.14 in the first equality for II we used independence of systems starting from different points, III T ; k, n; ℓ, m = 1 F 2 T Z T t Z Ts X j ∈Z E T r ϕρ j k,n ET r ′ ψρ j ℓ,m d r ′ d r = 1 F 2 T Z T t Z Ts Z R E T r ϕx + h k,n xET r ′ ψx + h ℓ,m xd x d r ′ d r. 3.15 Note that |h k,n x| ≤ 1, x ∈ R. 3.16 In each case we show convergence of I, II and III, thus proving 3.9. It is shown that I, II, III are bounded, so the passage to the limit in each sum in 3.12 is justified. Particle systems with quasi-homogeneous initial states and their occupation time fluctuations 199 Next, we show that 〈X t, ϕ〉 ⇒ 〈X t, ϕ〉, ϕ ∈ S R, t ≥ 0. To this end, by 3.8 and 3.9 it suffices to prove that the Lyapunov condition lim T →∞ X j ∈Z 1 F 3 T E Z T t 〈N j r , ϕ〉 − E〈N j r , ϕ〉d r 3 = 0 is satisfied, and this property follows if we show that lim T →∞ X j ∈Z 1 F 3 T E Z T t 〈N j r , ϕ〉d r 3 = 0, t ≥ 0, ϕ ∈ S R, ϕ ≥ 0. 3.17 It is clear that convergence in law of linear combinations P m k=1 a k 〈X T t k , ϕ k 〉 can be obtained analogously from 3.9 and 3.17, thus establishing the claimed convergence X T ⇒ f X . In order to give 3.17 a more tractable form we use 2.1, 3.7, and the trivial inequality a 1 + . . . + a k 3 ≤ 3k 2 a 3 1 + . . . + a 3 k , a 1 . . . , a k ≥ 0, obtaining X j ∈Z 1 F 3 T E Z T t 〈N j r , ϕ〉d r 3 = X j ∈Z 1 F 3 T ∞ X k=0 p k E k X n=1 Z T t 〈N ρ j k,n r , ϕ〉d r 3 ≤ 3 1 F 3 T ∞ X k=0 p k k 2 k X n=1 X j ∈Z E Z T t 〈N ρ j k,n r , ϕ〉d r 3 ≤ 3Eθ 3 sup n,k ∈Z+ n ≤k 1 F 3 T Z R E Z T t 〈N x+h k,n x r , ϕ〉d r 3 d x see 3.11. So, to prove 3.17 it suffices to show that lim T →∞ sup n,k ∈Z+ n ≤k 1 F 3 T Z R E Z T t 〈N x+h k,n x r , ϕ〉d r 3 d x = 0, t ≥ 0, ϕ ∈ S R, ϕ ≥ 0. 3.18

3.3 Proof of Theorem 2.2a