Some Limit Theorems for a Particle System of Single Point Catalytic ...

Acta Mathematica Sinica, English Series Jun., 2007, Vol. 23, No. 6, pp. 997–1012 Published online: Apr. 12, 2006 DOI: 10.1007/s10114-005-0757-4 Http://www.ActaMath.com

Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks Vladimir VATUTIN1) Steklov Mathematical Institute, Gubkin street, 8, 119991, Moscow, Russia E-mail: [email protected]

Jie XIONG2) Department of Mathematics, University of Tennessee, Knoxville, TN 37996–1300, USA and Department of Mathematics, Hebei Normal University, Shijiazhuang 050016, P. R. China E-mail: [email protected] Abstract We study the scaling limit for a catalytic branching particle system whose particles perform random walks on Z and can branch at 0 only. Varying the initial (finite) number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are nβ √ particles and consider the scaled process Ztn (•) = Znt ( n •), where Zt is the measure-valued process representing the original particle system. We prove that Ztn converges to 0 when β < 14 and to a 1 nondegenerate discrete distribution when β = 14 . In addition, if 14 < β < 12 then n−(2β− 2 ) Ztn converges to a random limit, while if β > 12 then n−β Ztn converges to a deterministic limit. Keywords Renewal equation, branching particle system, scaling limit MR(2000) Subject Classification 60J80, 60K25

1 Introduction Since the early work of Dawson and Fleischmann, catalytic superprocesses have been studied by many authors. We refer the reader to the survey papers of Dawson and Fleischmann [1] and Klenke [2] for an account of this development. Inspired by the continuous case, many authors have studied the catalytic super random walks (CSRW) (cf. Greven, et al. [3] and the references therein). In [4], Dawson and Fleischmann introduced and studied the catalytic super-Brownian motion (CSBM) with a single point catalyst. This process is also studied by Fleischmann and Le Gall [5] from another point of view. Kaj and Sagitov [6] considered the discrete counterpart of this process, i.e. the CSRW with a single point catalyst. They proved that, if the system starts √ with n number of particles at 0 (each with mass n−1/2 ), then, under Brownian scaling, the CSRW converges to a CSBM with a single point catalyst. The aim of this article is to study various other limiting behaviors of the CSRW under Brownian scaling. Now we introduce the CSRW model in more details. Consider a system of particles performing independent standard continuous-time random walks on Z. Namely, each particle stays at a site other than 0 for an exponential time, and then moves with probability 12 to the left and with probability 12 to the right. At its new position, it waits for another exponential time and moves on. The behavior of a particle at state 0 is a bit different. It stays here for an exponential Received April 21, 2004, Accepted January 19, 2005 1) Research supported partially by DFG and grants RFBR 02-01-00266 and Russian Scientific School 1758.2003.1 2) Research supported partially by NSA and by Alexander von Humboldt Foundation

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time and then either moves to the left or to the right or dies or splits into two particles. The particle selects each of these four options with probability 14 . All the exponential waiting times at each site for each particle are of parameter 1 and are independent of each other. In this case the offspring generating function is f (s) = 12 (1 + s2 ). The newborn particles appear at point 0, and move and branch in the same fashion as their parent. The model described is a particular case of the branching random walk in a catalytic medium. The long-term behavior of such processes with various types of catalytic media were studied by many authors (cf. Greven, et al. [3] and Wakolbinger [7] and the references therein). The long-time limit of the moments of the population size process for the current model of single point catalyst was considered by Albeverio and Bogachev [8], Albeverio, et al. [9], Bogachev and Yarovaya [10, 11]. Topchii and Vatutin [12], [13] and Topchii et al. [14] studied the limiting behavior for the population size of the process as well as the subpopulation size at the catalyst position. In this paper, we consider the long-term behavior of this system as an evolution of measures. ¯ at time t, where R ¯ is the one-point Let Zt (A) be the number of particles in the region A ⊂ R compactification of R. The aim of this article is to study the long-term behavior of Zt . Namely, √ we consider the scaling limit of Zt . We scale the time by a factor n and the space by n so that the scaled spatial motions of particles are approximated by Brownian motions and define √ ¯ be the space of finite Borel measures on the random measures Ztn (•) = Znt ( n •). Let MF (R) n ¯ ¯ R. Then Z is an MF (R)-valued process. Denote the Dirac measure at 0 by δ0 . Let τ be the branching time for a random walk particle starting at 0. Let G be the distribution of τ . Let τ0 , τ1 , τ2 ,. . ., be i.i.d. with common exponential distribution of parameter 1. Let η0 be the first time for a standard continuous-time random walk starting at 1 or −1 to hit 0. Let η1 , η2 ,. . ., be independent copies of η0 . Here τ1 , τ2 ,. . ., are the exponential times a particle spent at 0, while η1 , η2 ,. . ., are the time intervals which this particle spent away from 0. Then τ = τ0 + η1 + τ1 + η2 + · · · + ηN + τN , where N is a random variable independent of τi , ηi , i = 1, 2, . . . , and having geometric distribution 1 , k = 0, 1, . . . . The following fact (cf. [14]) will be used with parameter 12 : P(N = k) = 2k+1 frequently in this article: As t → ∞,   1 d2 1 − G0 (t) := P (η0 > t) = √ + o √ , (1.1) 2 t t where d > 0 is a constant known explicitly. First, we investigate how many initial individuals are needed for Ztn to have a nontrivial limit without extra scaling on the mass of particles. Throughout the paper we denote by ¯ Lipschitz continuous functions on R, and for a measure μ ∈ MF (R) φ1 , . . . , φk nonnegative  write μ, φ := φdμ. The Brownian meander plays an important role in the description of the limits in this paper. Let Wt be a standard Brownian motion defined on [0, 1] and let s0 be the last time in [0, 1] that W reaches 0. Then Wt+ := (1 − s0 )−1/2 |W (s0 + t(1 − s0 ))|, 0 ≤ t ≤ 1 is called the Brownian ˆ t , 0 ≤ t ≤ 1} be meander (cf. Iglehart [15] and Durrett, et al. [16] for its properties). Let {W + + 1 the process which equals {Wt , 0 ≤ t ≤ 1} with probability 2 and {−Wt , 0 ≤ t ≤ 1} with probability 12 . Theorem 1.1 i) If β < 14 and Z0n = nβ δ0 , then Ztn converges in probability to 0 as n → ∞, for any fixed t > 0. ii) If Z0n = n1/4 δ0 , then, for any 0 < t1 < · · · < tk < ∞, (Ztn1 , . . . , Ztnk ) converges in law on ¯ k as n → ∞ to a tuple of measures (Zt∞ , . . . , Zt∞ ) whose law is determined by MF (R) 1 k     k √  ∞ i E exp − Zti , φ = exp(− 2g(t1 , . . . , tk )), i=1

Catalytic Branching Random Walks

with 2

2

g (t1 , . . . , tk ) := d

k 

−1/2 ti E

i=1

999



 exp

−

i−1 

 √ ˆ √ −φi ( ti W 1) ˆ φ ( ti Wtj /ti ) (1 − e ) . j

(1.2)

j=1

Let us give a heuristic commentary to this result. Setting φj = λj , j = 1, . . . , k, with λj ≥ 0, we get

       k k   −1/2 − i−1 λj ∞ ¯ 2 −λ j=1 i λi Zti (R) = exp − 2d ti e (1 − e ) . H(λ1 , . . . , λk ) := E exp −



i=1

i=1

This shows that the limiting measures Zt∞ (·), i = 1, . . . , k are discrete. Moreover, one can i deduce from the results of [17] that in the limit only offsprings of a Poisson number of the initial particles survive. Besides, associating with the branching random walk we consider here a Bellman–Harris branching process (see, for instance, [12], [13] or [14]) and referring to a statement from [18] one can conclude that, if the population of our branching random walk generated by an individual at time zero is nonempty at a large moment T , then the surviving members of this population have a chance to visit the origin only at moments o(T ) (see Lemma 2.5 below for analogous arguments). Since the total size of the population in the limit is finite with probability 1, this means that all the individuals in the limiting population were “born” at moment t = 0 and then never returned described, say, by the measure Zt∞ i to the origin. Hence in the limit individuals surviving up to a (scaled) moment tk perform a motion according to a Brownian meander. Put now λj = 0, j < k and let λk → 0. Then  

  −1/2 −1/2 1 − H(0, . . . , 0, λk ) = 1 − exp − 2d2 tk λk ∼ d 2tk λk . ¯ belongs to the normal domain of Hence (see, for instance, [19]) the distribution of Zt∞ (R) k attraction of a one-sided stable law with index 1/2. Clearly, the same conclusion is valid for ¯ i = 1, . . . , k. In particular, this means that, if we have a collection of i.i.d. random (R), any Zt∞ i ¯ r = 1, 2, . . . , N, distributed like Z ∞ (R), ¯ then, as N → ∞, (R), variables of Zt∞ tk k ,r N ∞ ¯ r=1 Ztk ,r (R) d → Z, (1.3) N2 where Z is a random variable obeying a one-sided stable law with index 1/2. It is natural to guess that if N grows with n not too fast then something like (1.3) has to be true for the prelimiting variables in point ii) of Theorem 1.1 as well. Our results confirm this hypothesis. 1 Indeed, take β > 14 . Then nβ initial particles can be divided into N = nβ− 4 groups of independent copies of branching particle systems, each of which has starting measure Z0n = n1/4 δ0 . The following two theorems say that if β is not too large, then N −2 Ztn has, as n → ∞, a random limit. In particular, the (scaled) total size of the population converges to a random variable having a stable law with index 1/2 (compare with [17]). If, however, β is large, then n−β Ztn has a deterministic limit in complete agreement with the law of large numbers. Theorem 1.2 Suppose that 14 < β < 12 . If Z0n = nβ δ0 and α = 2β − 12 , then for any ¯ k as n → ∞ to a 0 < t1 < · · · < tk < ∞, (n−α Ztn1 , . . . , n−α Ztnk ) converges in law on MF (R) ∞ ∞ tuple of measures (Zt1 , . . . , Ztk ) whose law is determined by     k √  ∞ i Zti , φ E exp − = exp(− 2g0 (t1 , . . . , tk )) with

g02

2

(t1 , . . . , tk ) , := d

i=1 √ ˆ −1/2 E[φi ( ti W 1 )]. i=1 ti

k

Note that Kaj and Sagitov [6] proved that, if Z0n = n1/2 δ0 , then n−1/2 Z n converges as n → ∞ to a catalytic super Brownian motion with a single point catalyst at 0. The following

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theorem says that, when the number of initial points is large enough, a law of large number behavior holds for our model: Theorem 1.3 If Z0n = nβ δ0 with β > 12 , then, for any nonnegative Lipschitz continuous function φ on R,  √   ˆ 1) 1 t E φ( t − uW  n−β Ztn , φ → du, n → ∞, π 0 u(t − u) in probability. These three theorems will be proved in Sections 2, 3, 4, respectively. We shall use c for a constant which can vary from place to place. 2 Limit Theorem for Ztn without Mass Scaling In this section we investigate how many initial particles are needed in order to get a nontrivial limit without scaling the mass of the particles. It turns out that when the initial number of points is of order n1/4 , a nontrivial limit is obtained. When the initial number of points is of order nβ with β < 14 , the random measure Ztn tends to 0. However, first we establish a number of simple results related to the properties of the random variables τi , ηi and the process ξt , 0 ≤ t < ∞, the standard continuous time random walk on Z. Set G1 (t) := P(τ1 + η0 ≤ t), denote ∞  1 ∗k G2 (t) := G (t), 2k 1 k=1

where G∗k 1 is the k-th convolution of G1 with itself. Clearly, G2 (t) is a distribution function and we let ζ be a random variable such that G2 (t) = P(ζ ≤ t). Lemma 2.1

As t → ∞, 1 − G1 (t) = P(τ1 + η0 > t) ∼ P(η0 > t) ∼

Proof It is easy to see, by t ≥ t0 , (1 − ε)d2 √ ≤ 2 t = ≤

2 d√ . 2 t

(1.1), that, for any ε > 0, there exists t0 = t0 (ε) such that, for all P(η0 > t) ≤ P(τ1 + η0 > t)

P(τ1 + η0 > t; τ1 ≤ 3 ln t) + P(τ1 + η0 > t; τ1 > 3 ln t) P(η0 > t − 3 ln t) + P(τ1 > 3 ln t) (1 + ε)d2 1 √ ≤ + . 2 t − 3 ln t t3 Since ε > 0 is arbitrary, the lemma follows. Lemma 2.2

As t → ∞, 1 − G2 (t) = P(ζ > t) ∼ 2P(η0 > t) = 2(1 − G0 (t)) ∼

d2 √ . t

Proof This statement is a particular case of Theorem 3, Section 4, Chapter IV in [20] if one takes γ = 1/2 in the mentioned theorem. Let G(t) := P(τ ≤ t) be the lifelength distribution of particles of the process. Corollary 2.3 As t → ∞,   1 d2 (2.1) 1 − G (t) = √ + o √ . t t Proof It is not difficult to check that P(τ > t) = P(τ0 + ζ > t) (2.2) and, to complete the proof one should repeat the arguments used in the proof of Lemma 2.1. The following two lemmas relate the lifelength of a particle starting from point 0 at moment t = 0 with its position ξt at moment t > 0 : Lemma 2.4 As t → ∞, P(τ > t, ξt = 0) = o(P(τ > t)).

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1001

Proof Clearly, P(τ > t, ξt = 0) =

∞ 

P(τ > t, ξt = 0, N = k) = P(τ0 > t) +

k=0

∞  k=1

1 2k+1

P(Sk < t, Sk + τk > t),

where S0 := 0, Sk := τ0 + η1 + τ1 + η2 + · · · + ηk , k = 1, 2, . . . . Therefore,  t ∞  1 P(τ > t, ξt = 0) = P(τ0 > t) + P(τk > t − u)dP(Sk ≤ u) 2k+1 0 k=1  t ∞  1 = e−t + e−(t−u) dG∗k 1 (u) 2k+1 0 k=1  1 t −(t−u) = e−t + e dG2 (u). 2 0 Splitting the integral into two parts and recalling Lemma 2.2 we get  t  t−3 ln t  t −(t−u) −(t−u) e dG2 (u) = e dG2 (u) + e−(t−u) dG2 (u) 0

0

t−3 ln t

1 ≤ 3 + G2 (t) − G2 (t − 3 ln t) = o(1 − G2 (t)). t This estimate, clearly, implies the statement of the lemma. Let θt := t − sup{0 ≤ u ≤ t : ξu = 0} denote the time passed after the last visit of a particle to zero. The next lemma shows that, given {τ > t}, the last visit occurred “very long time” ago. As t → ∞, √ P(τ > t, θt < t − t) = o(P(τ > t)), t → ∞. Proof In view of the previous lemma it is enough to demonstrate that √ P(τ > t, 0 < θt < t − t) = o(P(τ > t)), t → ∞. Similarly to Lemma 2.4, ∞  √ √ P(τ > t, 0 < θt < t − t) = P(τ > t, 0 < θt < t − t, N = k)

Lemma 2.5

k=0 ∞ 

√ 1 P(S + τ < t, S > t, θ < t − t) k−1 k−1 k t 2k k=1  t ∞  1 = P(Sk−1 + τk−1 ∈ du, ηk > t − u). 2k √t =

k=1

Denoting, for brevity, G3 (t) := 1 − e−t , t ≥ 0, we have  √ 1 t (1 − G0 (t − u))e−u du P(τ > t, 0 < θt < t − t) = 2 √t  t ∞  1 (k−1)∗ + (1 − G0 (t − u))d(G3 ∗ G1 (u)) 2k √t k=2  1 t (1 − G0 (t − u))e−u du = 2 √t  1 t + √ (1 − G0 (t − u))d(G3 ∗ G2 (u)). 2 t Clearly,  t √ −u du ≤ e− t = o(P(τ > t)), t → ∞. √ (1 − G0 (t − u))e t

(2.3)

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Recalling (2.2) that G(t) = G3 ∗ G2 (t), we have, for any ε ∈ (0, 1)  t √ (1 − G0 (t − u))d(G3 ∗ G2 (u)) t

 =

(1−ε)t

√ t

 (1 − G0 (t − u))dG(u) +

t

(1−ε)t

(1 − G0 (t − u))dG(u)

√ ≤ (1 − G0 (tε))(1 − G( t)) + G(t) − G(t(1 − ε)). By (1.1) and (2.1) we see that √ (1 − G0 (tε))(1 − G( t)) + G(t) − G(t(1 − ε)) = 0. lim sup lim sup 1 − G(t) t→∞ ε→+0 Combining this fact with the preceding estimates we easily get (2.3). We have finished the preliminary estimates and are ready now to pass to the proofs of our first main statement, Theorem 1.1. For 0 ≤ t1 < · · · < tk < ∞, define     k   i ¯ := Eδ 1 − exp − Z . Qt¯(φ) , φ ti 0 i=1

Then

 Eδ0 1 − exp



 k   n i Zti , φ − = Qnt¯(φ¯n ), i=1

where φin (x) = φi (n−1/2 x). To prove Theorem 1.1,√we first prove lim n1/4 Qnt¯(φ¯n ) = tg(t1 , . . . , tk ). n→∞

(2.4)

Consider the case k = 1 and take a nonnegative Lipschitz continuous test function φ with φ (0) > 0. Let Qt (φ) = Eδ0 (1 − exp {− Zt , φ}) . The following renewal equation plays a pivotal role in the proofs of all results in this article:  t Qt (φ) = E0 [1 − exp{−φ (ξt )} | τ > t] (1 − G (t)) + (1 − f (1 − Qt−u (φ))) dG (u) . (2.5) 0

Set φn (x) = φ(n−1/2 x) and h (φn , nt) = E0 [1 − exp{−φn (ξnt )} | τ > nt] . It is easy to check that Eδ0 [1 − exp {− Ztn , φ}] = Qnt (φn ) satisfies the following scaled renewal equation:  nt Qnt (φn ) = h (φn , nt) (1 − G (nt)) + (1 − f (1 − Qnt−u (φn ))) dG (u) . (2.6) 0

Lemma 2.6

For t > 0, we have √ ˆ 1 )}], n → ∞. h (φn , nt) → E0 [1 − exp{−φ( tW (2.7) Proof Note that there is a Poisson random variable M = M (n, t) of parameter θnt such that ξnt − ξnt−θnt = S˜M , ˜ where Sk is the partial sum of an i.i.d. sequence of random variables of mean 0. The condition minnt−θnt ≤u≤nt ξu > 0 is equivalent to that the hitting time of S˜k for the set (−∞, 0] is greater than M . By a proposition in Iglehart [15], we have ξnt − ξnt−θnt √ as n → ∞. → W1+ , M √ M θnt Under the condition nt − θnt ≤ nt, we have M n = θnt n → t, a.s. Therefore, the conditional probability measure   √ ξnt − ξnt−θnt  √ ∈ ·  nt − θnt ≤ nt, min ξu > 0 P nt−θnt ≤u≤nt n √ + converges to the probability measure induced by tW1 as n → ∞. Evidently,   √ ξnt−θnt  √ nt − θ ≤ nt → 0, n → ∞,  nt n

Catalytic Branching Random Walks

in probability. As

 φn (ξnt ) = φ

we have

1003

ξnt − ξnt−θnt ξnt−θ √ + √ nt n n

 ,

  √  E0 1 − exp{−φn (ξnt )} θnt ≥ nt − nt,

 min ξu > 0 nt−θnt ≤u≤nt √ → E0 [1 − exp{−φ( tW1+ )}]. Similar arguments are valid for the case maxnt−θnt ≤u≤nt ξu < 0 (with replacement of W1+ by −W1+ ). Since each of these possibilities happens with probability 1/2, we see that √ √ ˆ 1 )}]. E0 [1 − exp{−φn (ξnt )} |θnt ≥ nt − nt, τ > nt] → E0 [1 − exp{−φ( tW

(2.7) then follows from Lemma 2.5. Our next lemma states that the main contribution of Qnt−u (φn ) to the integral in (2.6) comes from those with small u. Lemma 2.7 Let a(nt, n) be defined by the equality  n5/8 t a (nt, n) 1 2 (1 − f (1 − Qnt−u (φn ))) dG (u) , (2.8) + Qnt (φn ) = g (t) √ + √ n nt 0 and for fixed 0 < t0 < T < ∞, let a(n) := supt0 ≤t≤T | a (nt, n) |. Then limn→∞ a(n) = 0. Proof Recalling relation (2.7) we get   2 √ ˆ 1 )}] + o(1)) √d + o √1 h(φn , nt)(1 − G(nt)) = (E[1 − exp{−φ( tW n nt   1 1 = g 2 (t) √ + o √ , n → ∞. (2.9) n n By Theorem 1 of [14], we have c . (2.10) Qnt (φn ) ≤ P (Znt > 0) ≤ (nt + 1)1/4 Hence c 1 − f (1 − Qnt−u (φn )) ≤ Qnt−u (φn ) ≤ . 1/4 (nt − u + 1) Consequently, the integral term in (2.6) (taken over the range [n5/8 t, nt]) does not exceed  nt  nt  nt−n7/8 t dG (u) dG (u) dG (u) c = c + c . (2.11) 1/4 1/4 5/8 7/8 5/8 t (nt − u + 1) t (nt − u + 1) t n nt−n n (nt − u + 1)1/4 Since the integrand is bounded by 1, the first term on the right-hand side of (2.11) is bounded by   1 d2 d2 7/8 1 − G(nt − n t) − (1 − G (nt)) =  1/2 − (nt)1/2 + o (nt)1/2 nt − n7/8 t   1 =o , as n → ∞. n1/2 On the other hand, the integrand for the second term on the right-hand side of (2.11) is bounded by (n7/8 t)−1/4 , and hence, the respective term is estimated from above by  nt−n7/8 t 1 1 dG (u) ≤ (1 − G(n5/8 t)) n7/32 t1/4 n5/8 t n7/32 t1/4   c 1 1 = + o( ) n7/32 t1/4 n5/16 t1/2 n5/16   1 = o . n1/2

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Plugging back to (2.11) we find that the integral term in (2.6) over the range [n5/8 t, nt] is o (n−1/2 ) as n → ∞. The conclusion of the lemma then follows from (2.9) and (2.6). Now we define a function ε (nt, n) by the equality √   2 ε (nt, n) Qnt (φn ) = g (t) 1/4 1 + . (2.12) ln (nt + 1) n Our next lemma says that Qnt−u (φn ) in (2.8) is close to Qnt (φn ) for 0 ≤ u ≤ n5/8 t. Lemma 2.8 Let ε+ (n) := supt0 ≤t≤T ε (nt, n) and ε− (n) := inf t0 ≤t≤T ε (nt, n) . Then ε+ (n) and ε− (n) are finite. Furthermore, there exist functions r ± (n, u) such that, for fixed 0 < t0 < T < ∞,  −  |r (n, u)| + |r + (n, u)| = 0, lim sup sup sup (2.13) n→∞ t0 ≤t≤T 0≤u≤n5/8 t

and there exists a constant n0 (t0 , T ) such that, for all n > n0 (t0 , T ) and t0 < t < T, R− (nt, u; n) ≤ Qnt−u (φn ) ≤ R+ (nt, u; n), where √   2 ε± (n) r ± (n, u) . R± (nt, u; n) := g (t) 1/4 1 + + ln (nt + 1) n n1/2 Proof Since φ (0) > 0 and φ is continuous, it follows that inf g(t) > 0. t0 ≤t≤T

(2.14)

(2.15)

By (2.12) and (2.10) we have ε (nt, n) Qnt (φn ) 1/4 1 1+ = √ n n1/4 ≤ c. ≤ P (Znt > 0) √ (2.16) ln (nt + 1) 2g (t) 2g (t) It follows from (2.15) and (2.16) that ε+ (n) < ∞ for each n > 0. Since Qnt (φn ) > 0, we have ε(nt, n) > − ln(1 + nt). Thus, ε− (n) > −∞ for each n > 0. Recall that g(t) is defined in (1.2) such that  √ ˆ 1) ). t1/4 g(t) = d E(1 − e−φ( tW Note that the density of the Brownian meander Wt+ (cf. [15]) at t = 1 is given by  2 x p(x) = x exp − , x > 0. 2 It is easy to show that, for t0 ≤ s < t ≤ T , √

√

+

+

|Ee−φ( tW1 ) − Ee−φ( sW1 ) | ≤ c(t0 , T )|t − s|, where c(t0 , T ) is a finite constant. Then |t1/4 g(t) − s1/4 g(s)| ≤ c1 (t0 , T )|t − s|. By (2.15) and (2.17) we have, for 0 ≤ u ≤ n5/8 t,       1/4    1/4  1 nt − u  nt − u  t g (t) − nt − u  g   ≤ ct − n  = o n1/4 . n n Clearly,   1 1 1 = +o 1/4 (nt)1/4 n1/2 (nt − u) and

1 1 = ln (nt − u + 1) ln (nt + 1)

By (2.12) we have

 Qnt−u (φn ) = g

nt − u n

 √

2

n1/4

   1 . 1+o n1/2

  ε(nt − u, n) 1+ . ln(nt − u + 1)

(2.17)

(2.18)

(2.19)

(2.20)

(2.21)

Catalytic Branching Random Walks

1005

Using (2.18)–(2.20) in (2.21), we get

√     1 2 ε(nt − u, n) Qnt−u (φn ) = g(t) 1/4 1 + . +o ln(nt + 1) n n1/2 Hence the conclusions of the lemma follows easily. ε(nt,n) ≤ c and, therefore, ε+ (n) ≤ Remark 2.9 Since Qnt (φn ) ≤ c(nt)−1/4 , we have 1 + ln(nt+1) c ln n. This, in view of (2.14), implies R+ (nt, u; n) → 0, n → ∞. Clearly, √  √    2 2 ε− (n) ε+ (n) g (t) 1/4 1 + ≤ Qnt (φn ) ≤ g (t) 1/4 1 + . (2.22) ln (nt + 1) ln (nt + 1) n n Our aim is to show that ε+ (n) ≤0 (2.23) lim sup n→∞ ln(nt + 1)

and lim inf n→∞

ε− (n) ≥ 0. ln(nt + 1)

(2.24)

Since the function 1 − f (1 − x) = x − 12 x2 is monotone increasing in x ∈ (−∞, 1), we have, by Lemmas 2.7 and 2.8, for all sufficiently large n:  n5/8 t    a (nt, n) 1 1 − f 1 − R− (tn, u; n) dG (u) ≤ Qnt (φn ) g 2 (t) √ + √ + n nt 0  n5/8 t    a (nt, n) 1 2 1 − f 1 − R+ (tn, u; n) dG (u) . + ≤ g (t) √ + √ n nt 0 √ 1/4 Lemma 2.10 lim supn→∞ n Qnt (φn ) ≤ 2g(t). Proof Clearly, if lim supn→∞ ε+ (n) < ∞, then (2.23) is valid and hence Lemma 2.10 follows from (2.22). Thus, we assume that lim supn→∞ ε+ (n) = ∞. Let n0 = 1 and nj+1 = min{n > nj : ε+ (n) > ε+ (nj )}. Under our assumption limj→∞ ε+ (nj ) = ∞. Evidently, for each j = 1, 2, 3, . . . , there exists tj ∈ [t0 , T ] such that 1 ε (tj nj , nj ) ≥ ε+ (nj ) − . (2.25) nj It is easy to see that √    n5/8 tj  j 2 ε+ (nj ) 5/8 + R (nj tj , u; nj )dG (u) = g (tj ) 1/4 1 + G n j tj ln (nj tj + 1) 0 n j

+



1 1/2

nj

= g (tj )

0

5/8

nj

tj

r + (nj , u)dG (u)

√     1 2 ε+ (nj ) 1 + , + o 1/4 1/2 ln (nj tj + 1) nj nj

where we took into account (2.13), (2.14) and (2.1). By the same argument, we have    2  5/8 2 1 1 nj tj  + 1 ε+ (nj ) 2 R (nj tj , u; nj ) dG (u) = g (tj ) 1/2 1 + + o 1/2 . 2 0 ln (nj tj + 1) n n j

(2.26)

(2.27)

j

Recalling (2.25), we get √  √      2 2 ε (nj tj , nj ) ε+ (nj ) c Qnj tj φnj = g (tj ) 1/4 1 + ≥ g (tj ) 1/4 1 + − . ln (n t + 1) ln (n t + 1) n j j j j j nj nj

Vatutin V. and Xiong J.

1006

Therefore, by (2.8), (2.26) and (2.27), we have √   2 ε+ (nj ) c g (tj ) 1/4 1 + − ln (nj tj + 1) nj n j

√     2 1 a (nj tj , nj ) ε+ (nj ) ≤ g (tj ) 1/2 + + g (t ) 1 + + o j 1/4 1/2 ln (nj tj + 1) (nj tj )1/2 nj nj nj  2 1 ε+ (nj ) 2 . − g (tj ) 1/2 1 + ln (nj tj + 1) n 2

1

j

After cancellations and evident transformations, Lemma 2.7 implies   2  ε+ (nj ) 2ε+ (nj ) + g 2 (tj ) = o (1) , j → ∞. ln (nj tj + 1) ln (nj tj + 1) Since inf t0 ≤t≤T g 2 (t) ≥ c > 0, we get ε+ (nj ) = o (ln (nj tj + 1)) = o (ln nj ) , j → ∞. For nj ≤ n < nj+1 , we have ε+ (n) ≤ ε+ (nj ) and, consequently, + lim supn→∞ εln(n) n

ε+ (n) ln n

≤

ε+ (nj ) ln nj .

This implies

= 0 and, in particular, (2.23) is valid. Hence the statement of the lemma follows. ε− (n) Note that 0 ≤ 1 + ln(nt+1) ≤ c. Now similarly to Lemma 2.10 we have √ Lemma 2.11 lim inf n→∞ n1/4 Qnt (φn ) ≥ 2g(t). From Lemmas 2.10 and 2.11, it follows that √ Qnt (φn ) ∼ 2n−1/4 g (t) . (2.28) Thus, we have finished the proof of (2.4) for k = 1. Next, we consider finite-dimensional distributions. For simplicity of notation, we treat the case k = 2 only. The general case is similar to that at the end of the next section, where the notation is relatively simpler,  so we have   treated  the  general k there. Let Qt1 ,t2 (φ1 , φ2 ) = Eδ0 1 − exp − Zt1 , φ1 − Zt2 , φ2 . Then 1 2 Qt1 ,t2 (φ1 , φ2 ) = 1 − E[e−φ (ξt1 )−φ (ξt2 ) 1{τ >t2 } ] −E[e−φ

1

(ξt1 )

f (1 − Qt2 −τ (φ2 ))1{t1 nt2 } ] 1

− E[e−φn (ξnt1 ) f (1 − Qnt2 −τ (φ2n ))1{nt1 nt2 } ] 1

+ E[(1 − e−φn (ξnt1 ) )1{nt1 nt1 } ] 1

+ E[e−φn (ξnt1 ) (1 − f (1 − Qnt2 −τ (φ2n )))1{nt1 nt2 ] → E[e−φ and

1

1

√ ˆ ( t2 W t1 /t2 )

E[1 − e−φn (ξnt1 ) | τ > nt1 ] → E[1 − e−φ

1

(1 − e−φ

√ ˆ ( t1 W 1)

Similarly to the proof of Lemma 2.7, we get 1

E[e−φn (ξnt1 ) (1 − f (1 − Qnt2 −τ (φ2n )))1{nt1 0, there exists m such that μ(Km ) > o ) > 1 −  for the same m. Since 1 − . In particular, μ(Km+1 o o lim sup μn (Km+1 ) ≥ μ(Km+1 ) > 1 − , Qnt1 ,nt2 (φ1n , φ2n ) = g 2 (t1 , t2 )

n→∞

o there exists N such that, for any n ≥ N , μn (Km+1 ) ≥ μn (Km+1 ) > 1 − . Therefore, {μn } is tight in P(E). Clearly, μ is the only limit for this sequence. ¯ satisfies the conditions of Lemma 2.12. Lemma 2.13 E := MF (R) ¯ ∼ P(R) ¯ × R+ under the map ν → ( ν¯ , ν(R)). ¯ ¯ Therefore, MF (R) Proof Clearly, MF (R) ν(R) ¯ ¯ ¯ has a compactification P(R) × R+ . Let Km = {μ ∈ MF (R) : μ(R) ≤ m}. The conditions in Lemma 2.12 can be verified easily. Completion of Proof of Theorem 1.1 Recall that Qnt¯(φ¯n ) is given at the beginning of this section. Note that     k  n i 1/4 Zti , φ lim En1/4 δ0 exp − = lim (1 − Qnt¯(φ¯n ))n n→∞

i=1

n→∞

= lim exp(n1/4 log(1 − Qnt¯(φ¯n ))) n→∞

= lim exp(−n1/4 Qnt¯(φ¯n )) n→∞ √ (2.30) = exp(− 2g(t1 , . . . , tk )). ¯ k . Thus, there exists It is trivial that (Ztn1 , . . . , Ztnk ) is tight in the compactification of MF (R) n n , . . . , Zt∞ ). a subsequence nj → ∞, j → ∞, such that (Zt1j , . . . , Ztkj ) converges to a limit (Zt∞ 1 k n n ∞ ∞ By (2.30) the limit is unique and hence (Zt1 , . . . , Ztk ) → (Zt1 , . . . , Ztk ) in distribution on the ¯ k . Applying Lemmas 2.12 and 2.13, we see that (Z n , . . . , Z n ) → compactification of MF (R) t1 tk ∞ ∞ ¯ k . This proves point ii) of Theorem 1.1. Point i) of (Zt1 , . . . , Ztk ) in distribution on MF (R) Theorem 1.1 then follows easily. 3 Random Limit for Rescaled Processes This section is devoted to the proof of Theorem 1.2. The major steps of the subsequent proofs are similar to those of Section 2 and for this reason we indicate only those parts of the proofs

Vatutin V. and Xiong J.

1008

which are essentially different. First, we give an estimate for the population size. Let Rt (λ) := Eδ0 [1 − exp {−λ Zt , 1}] , λ ∈ [0, ∞). There exists a constant c such that, for all λ ∈ [0, ∞),   Rt (λ) ≤ c 1 − e−λ 1 − G(t).

Lemma 3.1

Proof By (2.5), we have −λ

Rt (λ) = (1 − e

 )(1 − G(t)) +

0

t

(1 − f (1 − Rt−u (λ))dG(u).

Applying the renewal theorem gives

 1 t 2 R (λ))dU 1 (u), 2 0 t−u ∞ where U 1 (t) = k=1 G∗k (t), and G∗k is the k-multiple convolution of G with itself. It is known (see, for instance, [20, Ch. IV, Section 3, Theorem 1]) that Rt (λ) is monotone decreasing in t, for each fixed λ > 0. Hence  t 2 Rt−u (λ)dU 1 (u) ≥ Rt2 (λ) U 1 (t) Rt (λ) = 1 − e−λ −

0

and, therefore, Rt (λ) ≤

 1 . 2(1 − e−λ )  U 1 (t)

By (2.1) and a statement in Section 14.3 of Feller [19], U 1 (t) (1 − G(t)) → π2 , t → ∞. From   this we get Rt (λ) ≤ c1 2(1 − e−λ ) 1 − G(t) as needed. Now we proceed to proving the convergence of the Laplace transform:    k √  −α n i  lim En1/4 δ0 exp − n Zti , φ (3.1) = exp(− 2g0 (t1 , . . . , tk )). n→∞

i=1

We start with the case k = 1. Let Q be the same as in Section 2. Then Eδ0 (1 − exp{−n−α Ztn , φ}) = Qnt (n−α φn ). Lemma 3.2 Let γ := α2 + 34 . Let b(nt, n) be defined by equality  nγ t  −α      −2β 2 −2β 1 − f 1 − Qnt−u n−α φn dG (u) . (3.2) Qnt n φn = n g0 (t) + b (nt, n) n + 0

For fixed 0 < t0 < T < ∞ let b(n) := supt0 ≤t≤T | b (nt, n) |. Then limn→∞ b(n) = 0. Proof Let h, G be as in the previous section. By (2.6) we have  nt  −α   −α      1 − f 1 − Qnt−u n−α φn dG (u) . (3.3) Qnt n φn = h n φn , nt (1 − G (nt)) + 0

Similarly to (2.7) we have, by means of Lemma 2.5 and Iglehart [15]), √   ˆ 1 )], n → ∞. nα h n−α φn , nt → E[φ( tW (3.4) −α −α Observe that Qs (n φn ) ≤ φ∞ n . Therefore, the integral term in (3.3) (taken over the range [nt(1 − ), nt]) does not exceed  nt c n−α dG (u) = cn−α ((1 − G(nt(1 − ))) − (1 − G(nt)) nt(1− )

By Lemma 3.1, we have

   1 1 1 − √ +o √ = cn−α  n nt nt(1 − ) c  √ √ = + o(n−(α+1/2) ). nα nt 1 −  + 1 −   Qu (n−α φn ) ≤ c n−α (1 − G(u)).

(3.5)

Catalytic Branching Random Walks

1009

Then the integral term in (3.3) (taken over the range [nγ t, nt(1 − )] ) does not exceed  nt(1− )  Qnt−u (n−α φn )dG(u) ≤ c n−α (1 − G(nt))(1 − G(nγ t)) nγ t

= = =

cn−α/2 (nt)−1/4 (nγ t)−1/2 cn−(α/2+1/4+γ/2) o(n−(α+1/2) ).

The conclusion of the lemma then follows from (2.1) and (3.2). Now we define a function ε0 (nt, n) by the equality   √ −β  −α  ε0 (nt, n) Qnt n φn = g0 (t) 2n 1+ . (3.6) ln (nt + 1) The following lemma says that Qnt−u (n−α φn ) in (3.3) is close to Qnt (n−α φn ) for 0 ≤ u ≤ nγ t. Lemma 3.3

Let − ε+ 0 (n) := sup ε0 (nt, n) and ε0 (n) := t0 ≤t≤T

inf

t0 ≤t≤T

ε0 (nt, n) .

− ± Then ε+ 0 (n) and ε0 (n) are finite. Furthermore, there exist functions r0 (n, u) such that, for all t ∈ [t0 , T ], we have  −  |r0 (n, u)| + |r0+ (n, u)| = 0, lim sup sup (3.7) n→∞ 0≤u≤nγ t

and there exists a constant n0 (t0 , T ) such that, for all n > n0 (t0 , T ),   R0− (nt, u, n) ≤ Qnt−u n−α φn ≤ R0+ (nt, u, n), where R0± (tn, u, n)

= g0 (t)

√

2n

−β

 1+

ε± 0 (n) ln (nt + 1)

 +

r0± (n, u) . nβ

Proof Since φ (0) > 0 and φ is continuous, it follows that inf t0 ≤t≤T g0 (t) > 0 and, in view Qnt (n−α φn ) ε0 (nt,n) of the inequality 1 + ln(nt+1) ≤ √2g (t) nβ , we conclude that ε+ 0 (n) < ∞ for each n > 0. 0

γ Similarly, ε− 0 (n) > −∞ for each n > 0. Similarly to (2.18), for 0 ≤ u ≤ n t, we have   1/4    1/4 nt − u  γ−1 t g0 (t) − nt − u g0 ).  = o(n  n n Note that, as n → ∞,   1 1 = + o n−β β β (nt) (nt − u)

and

   1 1 = 1 + o nγ−1 . ln (nt − u + 1) ln (nt + 1)

(3.8)

(3.9)

(3.10)

The conclusions of the lemma then follow easily. With these preparations, the proof of the following lemma is similar to that of Lemma 2.3. We omit the details. √ Lemma 3.4 limn→∞ nβ Qnt (nα φn ) = 2g0 (t). Now we consider the multi-dimensional case and introduce additional notations. For a fixed tuple 0 = t0 < t1 < · · · < tk < tk+1 = ∞ and Lipschitz continuous functions φ1 , . . . , φk , set ¯ := Qt ,..., t (φj , . . . , φk ), 1 ≤ j ≤ k, and Qt¯ ¯ :≡ 0. Qt¯ (φ) (φ) j,k

j

k

k,k+1

Vatutin V. and Xiong J.

1010

¯ := Qt −u,..., t −u (φj , . . . , φk ). Note that For a fixed u ∈ [0, tj ) we use the notation Qt¯j, k −u (φ) j k Qt1 ,..., tk (φ1 , . . . , φk )     k   ¯ = E 1 − exp − = Qt¯1,k (φ) Zti , φi =1−

k+1 

 E exp

−





j=1

=1−

k+1 

i=1



E

k  i=1

−

exp

   Zti , φi 1{tj−1