Competing super-Brownian motions as limits of interacting particle systems

We study two-type branching random walks in which the birth or death rate of each type can depend on the number of neighbors of the opposite type. This competing species model contains variants of Durrett's predator-prey model and Durrett and Levin's colicin model as special cases. We verify in some cases convergence of scaling limits of these models to a pair of super-Brownian motions interacting through their collision local times, constructed by Evans and Perkins.


Introduction
Consider the contact process on the fine lattice Z N ≡ Z d /( √ N M N ). Sites are either occupied by a particle or vacant.
• Particles die at rate N and give birth at rate N + θ • When a birth occurs at x the new particle is sent to a site y = x chosen at random from x + N where N = {z ∈ Z N : z ∞ ≤ 1/ √ N } is the set of neighbors of 0.
• If y is vacant a birth occurs there. Otherwise, no change occurs.
The √ N in the definition of Z N scales space to take care of the fact that we are running time at rate N . The M N serves to soften the interaction between a site and its neighbors so that we can get a nontrivial limit. From work of Bramson, Durrett, and Swindle (1989) it is known that one should take Mueller and Tribe (1995) studied the case d = 1 and showed that if we assign each particle mass 1/N and the initial conditions converge to a continuous limiting density u(x, 0), then the rescaled particle system converged to the stochastic PDE: where dW is a space-time White noise.  considered the case d ≥ 2. To state their result we need to introduce super-Brownian motion with branching rate b, diffusion coefficient σ 2 , and drift coefficient β. Let M F = M F (R d ) denote the space of finite measures on R d equipped with the topology of weak convergence. Let C ∞ b be the space of infinitely differentiable functions on R d with bounded partial derivatives of all orders. Then the above super-Brownian motion is the M F -valued process X t , which solves the following martingale problem: is a martingale with quadratic variation < Z(φ) > t = t 0 X s (bφ 2 ) ds. Durrett and Perkins showed that if the initial conditions converge to a nonatomic limit then the rescaled empirical measures, formed by assigning mass 1/N to each site occupied by the rescaled contact processes, converge to the super-Brownian motion with b = 2, σ 2 = 1/3, and β = θ − c d . Here c 2 = 3/2π and in d ≥ 3, c d = ∞ n=1 P (U n ∈ [−1, 1] d )/2 d with U n a random walk that takes steps uniform on [−1, 1] d . Note that the −u 2 interaction term in d = 1 becomes −c d u in d ≥ 2. This occurs because the environments seen by well separated particles in a small macroscopic ball are almost independent, so by the law of large numbers mass is lost due to collisions (births onto occupied sites) at a rate proportional to the amount of mass there.
There has been a considerable amount of work constructing measure-valued diffusions with interactions in which the parameters b, σ 2 and β in (1.1) depend on X and may involve one or more interacting populations. State dependent σ's, or more generally state dependent spatial motions, can be characterized and constructed as solutions of a strong equation driven by a historical Brownian motion (see Perkins (1992), (2002)), and characterized as solutions of a martingale problem for historical superprocesses (Perkins (1995)) or more simply by the natural extension of (1.1) (see Donnelly and Kurtz (1999) Donnelly and Kurtz (1999)). (Historical superprocesses refers to a measure-valued process in which all the genealogical histories of the current population are recorded in the form of a random measure on path space.) State dependent branching in general seems more challenging. Many of the simple uniqueness questions remain open although there has been some recent progress in the case of countable state spaces (Bass and Perkins (2004)). In Dawson and Perkins (1998) and Dawson et al (2002), a particular case of a pair of populations exhibiting local interaction through their branching rates (called mutually catalytic or symbiotic branching) is analyzed in detail thanks to a couple of special duality relations. State dependent drifts (β) which are not "singular" and can model changes in birth and death rates within one or between several populations can be analyzed through the Girsanov techniques introduced by Dawson (1978) (see also Ch. IV of Perkins (2002)). Perkins (1994,1998) study a pair of interacting measure-valued processes which compete locally for resources through an extension of (1.1) discussed below (see remark after Theorem 1.1). In two or three dimensions these interactions involve singular drifts β for which it is believed the change of measure methods cited above will not work. In 3 dimensions this is known to be the case (see Theorem 4.14 of Evans and Perkins (1994)). Corresponding models with infinite variance branching mechanisms and stable migration processes have been constructed by Fleischmann and Mytnik (2003).
Given this work on interacting continuum models, it is natural to consider limits of multitype particle systems. The simplest idea is to consider a contact process with two types of particles for which births can only occur on vacant sites and each site can support at most one particle. However, this leads to a boring limit: independent super-processes. This can be seen from Section 5 in  which shows that in the single type contact process "collisions between distant relatives can be ignored." To obtain an interesting interaction, we will follow Durrett and Levin (1998) and consider two types of particles that modify each other's death or birth rates. In order to concentrate on the new difficulties that come from the interaction, we will eliminate the restriction of at most one particle per site and let ξ i,N t (x) be the number of particles of type i at x at time t. Having changed from a contact process to a branching process, we do not need to let M N → ∞, so we will again simplify by considering the case M N ≡ M . Let σ 2 denote the variance of the uniform distribution on (Z/M ) ∩ [−1, 1].
Letting x + = max{0, x} and x − = max{0, −x}, the dynamics of our competing species model may be formulated as follows: • When a birth occurs, the new particle is of the same type as its parent and is born at the same site.
• For i = 1, 2, let n i (x) be the number of individuals of type i in x + N . Particles of type i give birth at rate N + γ + i 2 −d N d/2−1 n 3−i (x) and die at rate N + γ − i 2 −d N d/2−1 n 3−i (x).
Here 3 − i is the opposite type of particle. It is natural to think of the case in which γ 1 < 0 and γ 2 < 0 (resource competition), but in some cases the two species may have a synergistic effect: γ 1 > 0 and γ 2 > 0. Two important special cases that have been considered earlier are (a) the colicin model. γ 2 = 0. In Durrett and Levin's paper, γ 1 < 0, since one type of E. coli produced a chemical (colicin) that killed the other type. We will also consider the case in which γ 1 > 0 which we will call colicin.
(b) predator-prey model. γ 1 < 0 and γ 2 > 0. Here the prey 1's are eaten by the predator 2's which have increased birth rates when there is more food.
Two related example that fall outside of the current framework, but for which similar results should hold: (c) epidemic model. Here 1's are susceptible and 2's are infected. 1's and 2's are individually branching random walks. 2's infect 1's (and change them to 2's) at rate γ2 −d N d/2 n 2 (x), while 2's revert to being 1's at rate 1.
(d) voter model. One could also consider branching random walks in which individuals give birth to their own types but switch type at rates proportional to the number of neighbours of the opposite type.
The scaling N d/2−1 is chosen on the basis of the following heuristic argument. In a critical branching process that survives to time N there will be roughly N particles. In dimensions d ≥ 3 if we tile the integer lattice with cubes of side 1 there will be particles in roughly N of the N d/2 cubes within distance √ N of the origin. Thus there is probability 1/N d/2−1 of a cube containing a particle. To have an effect over the time interval [0, N ] a neighbor of the opposite type should produce changes at rate N −1 N d/2−1 or on the speeded up time scale at rate N d/2−1 . In d = 2 an occupied square has about log N particles so there will be particles in roughly N/(log N ) of the N squares within distance √ N of the origin. Thus there is probability 1/(log N ) of a square containing a particle, but when it does it contains log N particles. To have an effect interactions should produce changes at rate 1/N or on the speeded up time scale at rate 1 = N d/2−1 . In d = 1 there are roughly √ N particles in each interval [x, x + 1] so each particle should produce changes at rate N −1 N −1/2 or on the speeded up time scale at rate N −1/2 = N d/2−1 .
Our guess for the limit process comes from work of Perkins (1994, 1998) who studied some of the processes that will arise as a limit of our particle systems. We first need a concept that was introduced by Barlow, Evans, and Perkins (1991) for a class of measure-valued diffusions dominated by a pair of independent super-Brownian motions. Let (Y 1 , Y 2 ) be an M F 2 -valued process. Let p s (x) s ≥ 0 be the transition density function of Brownian motion with variance σ 2 s.
For any φ ∈ B b (R d ) (bounded Borel functions on R d ) and δ > 0, let The collision local time of (Y 1 , Y 2 ) (if it exists) is a continuous non-decreasing M F -valued stochastic process t → L t (Y 1 , Y 2 ) such that for all t > 0 and φ ∈ C b (R d ), the bounded continuous functions on R d . It is easy to see that if Y i s (dx) = y i s (x)dx for some Borel densities y i s which are uniformly bounded on compact time intervals, then L t (Y 1 , Y 2 )(dx) = t 0 y 1 s (x)y 2 s (x)dsdx. However, the random measures we will be dealing with will not have densities for d > 1.
The final ingredient we need to state our theorem is the assumption on our initial conditions.
Let B(x, r) = {w ∈ R d : |w − x| ≤ r}, where |z| is the L ∞ norm of z. For any 0 < δ < 2 ∧ d we set N δ (µ) ≡ inf : sup x µ(B(x, r)) ≤ r (2∧d)−δ , for all r ∈ [N −1/2 , 1] , where the lower bound on r is being dictated by the lattice Z d /( √ N M ). We say that a sequence of measures µ N , N ≥ 1 satisfies condition UB N if sup N ≥1 N δ (µ N ) < ∞, for all 0 < δ < 2 ∧ d We say that measure µ ∈ M F (R d ) satisfies condition UB if for all 0 < δ < 2 ∧ d δ (µ) ≡ inf : sup x µ(B(x, r)) ≤ r (2∧d)−δ , for all r ∈ (0, 1] < ∞ If S is a metric space, C S and D S are the space of continuous S-valued paths and càdlàg S-valued paths, respectively, the former with the topology of uniform convergence on compacts and the latter with the Skorokhod topology. C k b (R d ) denotes the set of functions in C b (R d ) whose partial derivatives of order k or less are also in C b (R d ).
The main result of the paper is the following. If X = (X 1 , X 2 ), let F X t denote the rightcontinuous filtration generated by X.
Theorem 1.1 Suppose d ≤ 3. Define measure-valued processes by Suppose γ 1 ≤ 0 and γ 2 ∈ R. If {X i,N 0 }, i = 1, 2 satisfy UB N and converge to X i 0 in M F for i = 1, 2, then {(X 1,N , X 2,N ), N ≥ 1} is tight on D M F 2 . Each limit point (X 1 , X 2 ) ∈ C M F 2 and satisfies the following martingale problem M γ 1 ,γ 2 Theorem 1.2 Let γ 1 = 0 and γ 2 ∈ R, d ≤ 3, and X i 0 , i = 1, 2, satisfy condition UB. Then there is a unique in law solution to the martingale problem (MP γ 1 ,γ 2 The uniqueness for γ 1 = 0 and γ 2 ≤ 0 above was proved by Evans and Perkins (1994) (Theorem 4.9) who showed that the law is the natural one: X 1 is a super-Brownian motion and conditional on X 1 , X 2 is the law of a super ξ-process where ξ is Brownian motion killed according to an inhomogeneous additive functional with Revuz measure X 1 s (dx)ds. We prove the uniqueness for γ 2 > 0 in Section 5 below. Here X 1 is a super-Brownian motion and conditional on X 1 , X 2 is the superprocess in which there is additional birthing according to the inhomogeneous additive functional with Revuz measure X 1 s (dx)ds. Such superprocesses are special cases of those studied by Dynkin (1994) and Kuznetsov (1994) although it will take a bit of work to connect their processes with our martingale problem.
Hence as an (almost) immediate Corollary to the above theorems we have: Proof We only need point out that by elementary properties of weak convergence X i 0 will satisfy UB since {X i,N 0 } satisfies UB N . The result now follows from the above three Theorems.
For d = 1 uniqueness of solutions to (1.3) for γ i ≤ 0 and with initial conditions satisfying (this is clearly weaker that each X 1 0 satisfying UB) is proved in Evans and Perkins (1994) (Theorem 3.9). In this case solutions can be bounded above by a pair of independent super-Brownian motions (as in Theorem 5.1 of Barlow, Evans and Perkins (1991)) from which one can readily see that In this case u 1 , u 2 are also the unique in law solution of the stochastic partial differential equation where W 1 and W 2 are independent white noises. (See Proposition IV.2.3 of Perkins (2002).) Turning next to γ 2 > 0 in one dimension we have the following result: Theorem 1.5 Assume γ 1 ≤ 0 ≤ γ 2 , X 1 0 ∈ M F has a continuous density on compact support and X 2 0 satisfies Condition UB. Then for d = 1 there is a unique in law solution to M γ 1 ,γ 2 X 1 0 ,X 2 0 which is absolutely continuous to the law of the pair of super-Brownian motions satisfying M 0,0 In particular X i (t, dx) = u i (t, x)dx for u i : (0, ∞) → C K continuous maps taking values in the space of continuous functions on R with compact support, i = 1, 2.
We will prove this result in Section 5 using Dawson's Girsanov Theorem (see Theorem IV. 1.6 (a) of Perkins (2002)). We have not attempted to find optimal conditions on the initial measures. As before, the following convergence theorem is then immediate from Theorem 1.1 0 has a continuous density with compact support. If X i,N are as in Theorem 1.1, then (X 1,N , X 2,N ) converges weakly in D M F 2 to the unique solution of MP γ 1 ,γ 2 Having stated our results, the natural next question is: What can be said about uniqueness in other cases? Conjecture 1.7 Uniqueness holds in d = 2, 3 for any γ 1 , γ 2 .
For γ i ≤ 0 Evans and Perkins (1998) prove uniqueness of the historical martingale problem associated with (1.3). The particle systems come with an associated historical process as one simple puts mass N −1 on the path leading up to the current position of each particle at time t. It should be possible to prove tightness of these historical processes and show each limit point satisfies the above historical martingale problem. It would then follow that in fact one has convergence of empirical measures in Theorem 1.1 (for γ i ≤ 0) to the natural projection of the unique solution to the historical martingale problem onto the space of continuous measure-valued processes.
In addition to expanding the values of γ that can be covered, there is also the problem of considering more general approximating processes. Conjecture 1.9 Our results hold for the long-range contact process with modified birth and death rates.
Returning to what we know, our final task in this Introduction is to outline the proofs of Theorems 1.1 and 1.2. Suppose γ 1 ≤ 0 and γ 2 ∈ R, and setγ 1 = 0 andγ 2 = γ + 2 . Proposition 2.2 below will show that the corresponding measure-valued processes can be constructed on the same space so that X i,N ≤X i,N for i = 1, 2. Here (X 1,N ,X 2,N ) are the sequence of processes corresponding to parameter values (γ 1 ,γ 2 ). Tightness of our original sequence of processes then easily reduces to tightness of this sequence of bounding processes, because increasing the measures will both increase the mass far away (compact containment) and also increase the time variation in the integrals of test functions with respect to these measure-valued processes-see the approximating martingale problem (2.12) below. Turning now to (X 1,N ,X 2,N ), we first note that the tightness of the first coordinate (and convergence to super-Brownian motion) is well-known so let us focus on the second. The first key ingredient we will need is a bound on the mean measure, including of course its total mass. We will do this by conditioning on the branching environmentX 1,N . The starting point here will be the Feynman-Kac formula for this conditional mean measure given below in (2.17). In order to handle tightness of the discrete collision measure forX 2,N we will need a concentration inequality for the rescaled branching random walkX 1,N , i.e., a uniform bound on the mass in small balls. A more precise result was given for super-Brownian motion in Theorem 4.7 of Barlow, Evans and Perkins (1991). The result we need is stated below as Proposition 2.4 and proved in Section 6.
Once tightness of (X 1,N , X 2,N ) is established it is not hard to see that the limit points satisfy a martingale problem similar to our target, (1.3), but with some increasing continuous measurevalued process A in place of the collision local time. To identify A with the collision local time of the limits, we take limits in a Tanaka formula for the approximating discrete local times (Section 4 below) and derive the Tanaka formula for the limiting collision local time. As this will involve a number of singular integrals with respect to our random measures, the concentration inequality forX 1,N will again play an important role. This is reminiscent of the approach in Evans and Perkins (1994) to prove the existence of solutions to the limiting martingale problem when γ i ≤ 0. However the discrete setting here is a bit more involved, since requires checking the convergence of integrals of discrete Green functions with respect to the random mesures. The case of γ 2 > 0 forces a different approach as we have not been able to derive a concentration inequality for this process and so must proceed by calculation of second moments-Lemma 2.3 below is the starting point here. The Tanaka formula derived in Section 5 (see Remark 5.2) is new in this setting.
Theorem 1.2 is proved in Section 5 by using the conditional martingale problem of X 2 given X 1 to describe the Laplace functional of X 2 given X 1 in terms of an associated nonlinear equation involving a random semigroup depending on X 1 . The latter shows that conditional on X 1 , X 2 is a superprocess with immigration given by the collision local time of a Brownian path in the random field X 1 .
Convention As our results only hold for d ≤ 3, we will assume d ≤ 3 throughout the rest of this work.

The Rescaled Particle System-Construction and Basic Properties
We first will write down a more precise description corresponding to the per particle birth and death rates used in the previous section to define our rescaled interacting particle systems. We let p N denote the uniform distribution on N , that is which the righthand side is absolutely summable. Set M = (M + (1/2)) d . The per particle rates in Section 1 lead to a process (ξ 1 , ξ 2 ) ∈ The factors of (M ) d may look odd but they combine with the kernels p N to get the factors of 2 −d in our interactive birth and death rates. Such a process can be constructed as the unique solution of an SDE driven by a family of Poisson point processes. For x, y ∈ Z N , let Λ Poisson processes on R 2 + , R 2 + , R 3 + , R 3 + , and R 2 + , respectively. Here Λ i,± x governs the birth and death rates at x, Λ i,±,c x,y governs the additional birthing or killing at x due to the influence of the other type at y and Λ i,m x,y governs the migration of particles from y to x. The rates of Λ i,± x are N ds du; the rates of Λ i,±,c x,y are N d/2−1 M p N (y − x)ds du dv; the rates of Λ i,m x,y are N p N (x − y)du. Let F t be the canonical right continuous filtration generated by this family of point processes and let F i t denote the corresponding filtrations generated by the point processes with superscript i, , 2-denote this set of initial conditions by S F -and consider the following system of stochastic jump equations for i = 1, 2, x ∈ Z N and t ≥ 0: Assuming for now that there is a unique solution to this system of equations, the reader can easily check that the solution does indeed have the jump rates described above. These equations are similar to corresponding systems studied in Mueller and Tribe (1994), but for completeness we will now show that (2.2) has a unique F t -adapted S F -valued solution. Associated with (2.2) introduce the increasing F t -adapted Z + ∪ {∞}-valued process x,y (ds, du).
Set T 0 = 0 and let T 1 be the first jump time of J. This is well-defined as any solution to (2.2) cannot jump until T 1 and so the solution is identically (ξ 1 0 , ξ 2 0 ) until T 1 . Therefore a short calculation shows that T 1 is exponential with rate at most At time T 1 (2.2) prescribes a unique single jump at a single site for any solution ξ and J increases by 1. Now proceed inductively, letting T n be the nth jump time of J. Clearly the solution ξ to (2.2) is unique up until T ∞ = lim n T n . Moreover Finally note that (2.3) and the corresponding bounds for the rates of subsequent times shows that J is stochastically dominated by a pure birth process starting at |ξ 1 0 | + |ξ 2 0 | and with per particle birth rate 4N + M N d/2−1 |(|γ 1 | + |γ 2 |). Such a process cannot explode and in fact has finite pth moments for all p > 0 (see Ex. 6.8.4 in Grimmett and Stirzaker (2001)). Therefore T ∞ = ∞ a.s. and we have proved (use (2.4) to get the moments below): Proposition 2.1 For each ξ 0 ∈ S F , there is a unique F t -adapted solution (ξ 1 , ξ 2 ) to (2.2). Moreover this process has càdlàg S F -valued paths and satisfies The following "Domination Principle" will play an important role in this work.
Proof. Let J and T n be as in the previous proof but forξ. One then argues inductively on n that ξ i t ≤ξ i t for t ≤ T n . Assuming the result for n (n = 0 holds by our assumption on the initial conditions), then clearly neither process can jump until T n+1 . To extend the comparison to T n+1 we only need consider the cases where ξ i jumps upward at a single site x for which ξ i T n+1 − (x) =ξ i T n+1 − (x) orξ i jumps downward at a single site x for which the same equality holds. As only one type and one site can change at any given time we may assume the processes do not change in any other coordinates. It is now a simple matter to analyze these cases using (2.2) and show that in either case the other process (the one not assumed to jump) must in fact mirror the jump taken by the jumping process and so the inequality is maintained at T n+1 . As we know T n → ∞ a.s. this completes the proof.
Denote dependence on N by letting ξ N = (ξ 1,N , ξ 2,N ) be the unique solution to (2.2) with a given initial condition ξ N 0 and let denote the associated pair of empirical measures, each taking values in M F . We will not be able to deal with the case of symbiotic systems where both γ i > 0 so we will assume from now on that γ 1 ≤ 0. As we prefer to write positive parameters we will in fact replace γ 1 with −γ 1 and therefore assume γ 1 ≥ 0. We will letξ i,N andX i,N denote the corresponding particle system and empirical measures withγ = (0, γ + 2 ). We call (X 1,N , X 2,N ) a positive colicin process, asX 1,N is just a rescaled branching random walk which has a non-negative local influence onX 2,N . The above Domination Principle implies In order to obtain the desired limiting martingale problem we will need to use a bit of jump calculus to derive the martingale properties of X i,N . Define the discrete collision local time for We denote the corresponding quantity for our bounding positive colicin process byL i,N . These integrals all have finite means by (2.5) and, in particular, are a.s. finite. LetΛ denote the predictable compensator of a Poisson point process Λ and letΛ = Λ −Λ denote the associated martingale measure. If ψ i : R + × Ω × Z N → R are F t -predictable define a discrete inner product by To deal with the convergence of the above sum note that its predictable square function is If ψ is bounded, the above is easily seen to be square integrable by (2.5), and so M i,N (ψ i ) t is an L 2 F t -martingale. More generally whenever the above expression is a.s. finite for all t > 0, M i,N t (ψ) is an F t -local martingale. The last two terms are minor error terms. We writeM i,N for the corresponding martingale measures for our dominating positive colicin processes.
Let ∆ N be the generator of the "motion" process B N · which takes steps according to p N at rate N : Let Π N s,x be the law of this process which starts from x at time s. We will adopt the convention Π N x = Π N 0,x . It follows from Lemma 2.6 of Cox, Durrett and Perkins (2000) that if σ 2 is as defined in Section 1 then for It is now fairly straightforward to multiply (2.2) by φ i (t, x)/N , sum over x, and integrate by parts to see that (X 1,N , X 2,N ) satisfies the following martingale problem M N,γ 1 ,γ 2 To derive the conditional mean ofX 2,N givenX 1,N we first note thatξ 1,N is in fact F 1 t -adapted as the equations forξ 1,N are autonomous sinceγ 1 = 0 and so the pathwise unique solution will be adapted to the smaller filtration. Note also that ifF t = F 1 ∞ ∨ F 2 t , thenΛ 2,± ,Λ 2,±,c ,Λ 2,m are all F t -martingale measures and soM 2,N (ψ) will be aF t -martingale whenever ψ : [0, T ] × Ω × Z N → R is bounded andF t -predictable. Therefore if ψ,ψ : [0, T ] × Ω × Z N → R are bounded, continuous in the first and third variables for a.a. choices of the second, andF t -predictable in the first two variables for each point in Z N , then we can repeat the derivation of the martingale problem for (X 1,N , X 2,N ) and see that whereM 2,N t (ψ) is now anF t -local martingale because the right-hand side of (2.10) is a.s. finite for all t > 0.
Fix t > 0, and a map φ : Z N × Ω → R which is F 1 ∞ -measurable in the second variable and satisfies One can check that ψ s , s ≤ t is given by which indeed does satisfy the above conditions on ψ. Therefore for ψ, φ as abovē It will also be convenient to use (2.15) to prove a corresponding result for conditional second moments.
Here p (n) x (s 1 , . . . , s n , t, y 1 , . . . , y n , φ) We now state the concentration inequality for our rescaled branching random walksX 1,N which will play a central role in our proofs. The proof is given in Section 6.
Proposition 2.4 Assume that the non-random initial measure Then for any δ > 0, H δ,N is bounded in probability uniformly in N , that is, for any > 0, there exists M ( ) such that P (H δ,N ≥ M ( )) ≤ , ∀N ≥ 1.
Throughout the rest of the paper we will assume It follows from (M N,0,0 ) and the above assumption that sup sX 1,N s (1) is bounded in probability uniformly in N . For example, it is a non-negative martingale with mean X 1,N 0 (1) → X 1 0 (1) and so one can apply the weak L 1 inequality for non-negative martingales. It therefore follows from Proposition 2.4 that (suppressing dependence on δ > 0) is also bounded in probability uniformly in N , that is The next two Sections will deal with the issues of tightness and Tanaka's formula, respectively. In the course of the proofs we will use some technical Lemmas which will be proved in Sections 7 and 8, and will involve a non-decreasing σ(X 1,N )-measurable processR N (t, ω) whose definition (value) may change from line to line and which also satisfies

Tightness of the Approximating Systems
It will be convenient in Section 4 to also work with the symmetric collision local time defined by This section is devoted to the proof of the following proposition.
where M i are continuous local martingales such that As has alreeay been noted, the main step will be to establish Proposition 3.1 for the positive colicin process (X 1,N ,X 2,N ) which bounds (X 1,N , X 2,N ). Recall that this process solves the following martingale problem: For bounded φ : Z N → R, Proposition 3.2 The sequence {X 1,N , N ≥ 1} converges weakly in D M F to super-Brownian motion with parameters b = 2, σ 2 , and β = 0.
Proof This result is standard in the super-Brownian motion theory, see e.g. Theorem 15.1 of Cox, Durrett, and Perkins (1999).
Most of the rest of this section is devoted to the proof of the following proposition.
Recall the following lemma (Lemmas 2.7, 2.8 of ) which gives conditions for tightness of a sequence of measure-valued processes.  (i) For each T, > 0, there is a compact set K T, ⊂ R d such that , and (iii), and for each φ ∈ Φ all limit points of P N,φ are supported by C R , then P N is tight in D M F and all limit points are supported on C M F .
Notation. We choose the following constants: 0 < δ <δ < 1/6, (3.5) and define Recall our convention with respect toR N (t, ω) from the end of Section 2. The proof of the following bound on the semigroup P g N s,t is deferred until Section 7.
As simple consequences of the above we have the following bounds on the conditional mean measures ofX 2,N .
The next lemma gives a bound for a particular test function φ and is essential for bounding the first moments of the approximate local times.
Proof Deferred to Section 7.
Lemma 3.8 For any > 0, T > 0, there exist r 1 such that Clearly, By (3.9) and Assumption 2.5 we get that for all r sufficiently large and all N ∈ N, Arguing in a similar manner for I 2,N , we get Again, by (3.9), our assumptions on {X 2,N 0 , N ≥ 1} and tightness of {R N (T ), N ≥ 1} and {X 1,N , N ≥ 1} we get that for all r sufficiently large and all N , and we are done.
Lemma 3.9 For any , 1 > 0, T > 0, there exists r 1 such that Now recalling Assumption 2.5 and the tightness of {R N (T ), N ≥ 1} and {X 1,N , N ≥ 1}, we complete the proof as in Lemma 3.8.
Proof Apply Chebychev's inequality on each term of the martingale problem (3.2) forX 2,N and then Doob's inequality to get Hence by tightness of {X 2,N 0 }, (3.4) and Lemmas 3.8, 3.9 we may take r sufficiently large such that the right-hand side of (3.14) is less than /2 with probability at least 1 − /2 for all N . This completes the proof.
where the last inequality follows by Lemma 3.7(b) with µ N =X 1,N s . As we may assume (the left-hand side is bounded in probability uniformly in N by Propostions 2.4 and 3.2), the result follows.
Lemma 3.12 For any T > 0, Proof Applying Chebychev's inequality on each term of the martingale problem (3.2) forX 2,N and then Doob's inequality, one sees that Now apply Assumption 2.5, (3.4), Lemma 3.11, and Corollary 3.6(b) to finish the proof.
Proof Lemmas 3.9 and 3.11 imply conditions (i) and (ii), respectively in Lemma 3.4. Now let us check (iii). Let Φ ⊂ C b (R d ) be a separating class of functions. We will argue by Aldous' tightness criterion (see Theorem 6.8 of Walsh (1986)). First by (2.22) and Lemma 3.11 we immediately get that for any φ ∈ Φ, t ≥ 0, {L 2,N t (φ) : N ∈ N} is tight. Next, let {τ N } be arbitrary sequence of stopping times bounded by some T > 0 and let { N , N ≥ 1} be a sequence such that N ↓ 0 as N → ∞. Then arguing as in Lemma 3.11 it is easy to verify Then by (2.22) and Lemma 3.12 we immediately get that in probability as N → ∞. Hence by Aldous' criterion for tightness we get that {L 2,N (φ)} is tight in D R for any φ ∈ Φ. NoteL 2,N (φ) ∈ C R for all N , and so {L 2,N (φ)} is tight in C R , and we are done.
The next lemma will be used for the proof of Proposition 3.1. The processes X i,N , L i,N and L N are all as in that result.
Lemma 3.14 The sequences {L i,N , N ≥ 1}, i = 1, 2, and {L N , N ≥ 1} are tight in C M F , and moreover for any uniformly continuous function φ on R d and T > 0 whereL 2,N is the approximating collision local time for the (X 1,N ,X 2,N ) solving M is tight in C M F , and hence, by (3.18), L 2,N is tight as well (see the proof of Lemma 3.13).
To finish the proof it is enough to show that for any uniformly continuous function φ on R d and T > 0 We will check only (3.19), since the proof of (3.20) goes along the same lines. By trivial calculations we get where the last inequality follows by (3.18). The result follows by the uniform continuity assumption on φ and Lemma 3.13. Now we are ready to present the Proof of Proposition 3.3 We will check conditions (i)-(iii) of Lemma 3.4. By Lemmas 3.10 and 3.12, conditions (i) and (ii) of Lemma 3.4 are satisfied. Turning to (iii), fix a φ ∈ C 3 b (R d ). Then using the Aldous criterion for tightness along with Lemma 3.12 and (2.11), and arguing as in Lemma 3.13, it is easy to verify that { · 0X 2,N s (∆ N φ) ds} is a tight sequence of processes in C R . By Lemma 3.13 and the uniform convergence of P N φ to φ we also see that Turning now to the local martingale term in (3.2), arguing as above, now using |∇φ| 2 ≤ C φ < ∞ and Lemma 3.13 as well, we see from (3.4) that { M 2,N (φ) · , N ≥ 1} is a tight sequence of processes in C R . Note also that by definition, Theorem VI.4.13 and Proposition VI.3.26 of Jacod and Shiryaev Jacod and Shiryaev (1987) show that {M 2,N above results with (3.2) and Corollary VI.3.33 of Jacod and Shiryaev Jacod and Shiryaev (1987) show thatX 2,N · (φ) is tight in D R and all the limit points are supported in C R . Lemma 3.4(b) now completes the proof.
Proof of Proposition 3.1 Arguing as in the proof of Proposition 3.3 and using Proposition 2.2 and Lemma 3.14, we can easily show that {(X 1,N , X 2,N , L 2,N , L 1,N , L N ), N ≥ 1} is tight on D M F 5 , and any limit point belongs to as k → ∞. By Skorohod's theorem, we may assume that convergence in (3.21) is a.s. in D M F 5 to a continuous limit. To complete the proof we need to show that (X 1 , X 2 ) satisfies the martingale , except perhaps the local martingale terms. By convergence of the other terms in M N k ,γ 1 ,γ 2 These local martingales have jumps bounded by 2 N k φ i ∞ , and square functions which are bounded in probability uniformly in N k by Proposition 3.2 and Lemma 3.12. Therefore they are locally bounded using stopping times {T N k n } which become large in probability as n → ∞ uniformly in N k . One can now proceed in a standard manner (see, e.g. the proofs of Lemma 2.10 and Proposition 2 in ) to show that M i (φ) have the local martingale property and square functions claimed in M −γ 1 ,γ 2 ,A apply the martingale problem with P δ φ i (P δ is the Brownian semigroup) and let δ → 0. As P δ ∆φ i → ∆φ i in the bounded pointwise sense, we do get M −γ 1 ,γ 2 ,A X 1 0 ,X 2 0 for φ i in the limit and so the proof is complete.

Convergence of the approximating Tanaka formulae
where α ≥ 0 for d = 3 and α > 0 for d ≤ 2. These conditions on α will be implicitly assumed in what follows. Note that for any bounded φ we have where p N · is the transition probability function of the continuous time random walk B N with generator ∆ N .
For 0 < < 1, define We introduce Then arguing as in Lemma 5.2 of Barlow, Evans, and Perkins (1991) where an Ito's formula for a pair of interacting super-Brownian motions was derived, we can easily verify the following approximate Tanaka formula for φ : Z → R bounded: be an arbitrary limit point of (X 1,N , X 2,N , L 1,N , L 2,N , L N ) (they exist by Proposition 3.1)), and to simplify the notation we assume as N → ∞. Moreover, throughout this section, by the Skorohod representation theorem, we may assume that A change of variables shows this agrees with the definition of G α φ in Section 5 of Barlow-Evans and Perkins (1991) and so is finite (and the above limit exists) for all x 1 = x 2 , and all ( Barlow, Evans and Perkins (1991)).
In this section we intend to prove the following proposition.
Proposition 4.1 Let (X 1 , X 2 , A) be an arbitrary limiting point described above. Then for φ ∈ C b (R d ), To verify the proposition we will establish the convergence of all the terms in (4.5) through a series of lemmas.
If µ ∈ M F (Z N ), µ * p N (dx) denotes the convolution measure on Z N . The proof of the following lemma is trivial and hence is omitted.
Now let us formulate a number of helpful results whose proofs are deferred to Section 8.
uniformly on the compact subsets of R d × R d .
Proof: Let ε 0 ∈ (0, 1). The key step will be to show Once (4.12) is established we see that the contribution to G α N φ from times s ≤ ε is small uniformly for |x 1 − x 2 | ≥ ε 0 and N . A straightforward application of the continuous time local central limit theorem (it is easy to check that (5.2) in Durrett (2004) works in continuous time) and Donsker's Theorem shows that uniformly in x 1 , x 2 in compacts, This immediately implies (b), and, together with (4.12), also gives (a). It remains to establish (4.12). Assume |x| ≡ |x 1 − x 2 | ≥ ε 0 and let {S j } be as in Lemma 7.1.
Now use Stirling's formula to conclude that for j ≥ 1 and ε = 2eε, Use this to bound (4.13) by Choose ε = ε(ε 0 ) such that the right-hand side is at most ε 0 for N ≥ N 0 (ε 0 ). By making ε smaller still we can handle the finitely many values of N ≤ N 0 and hence prove (4.12).
Lemma 4.9 For any φ ∈ C b,+ (R d ), T > 0, Proof Let f ∈ [0, 1] be a continuous function on R d satisfying (4.18). Then With Lemma 4.6(a) at hand, it is easy to check that for any compact Therefore by the convergence (X 1,N , L 2,N ) → (X 1 , A), in D M F 2 , P − a.s.
Lemma 4.11 For any φ ∈ C b,+ (R d ) and T > 0, Proof The proof goes along the same lines as of Lemma 4.10, with the only difference being that we use Lemmas 4.3 and 4.2 instead of Lemma 4.5.
Before we formulate the next lemma, let us introduce the following notation for the martingales in the approximate Tanaka formula (4.5): Proof Lemmas 4.7, 4.8 and 4.9 show that all the terms in (4.5), except perhapsM N t (φ), converge in probability, uniformly for t in compact time sets, to an a.s. continuous limit as N → ∞. Hence there is an a.s. continuous F X,A t -adapted processM t (φ) such that sup t≤T |M N t (φ) −M t (φ)| → 0 in probability as N → ∞.
(4.2) and Lemma 7.2 below imply In view of |∆X i,N s (1)| ≤ N −1 , and Proposition 3.1, we see that if it is easy and standard to check thatM (φ) is a continuous F X,A t -local martingale.
Proof of Proposition 4.1 Immediate from the approximate Tanaka formula (4.5), the Lemmas 4.7, 4.8, 4.9, 4.12 and convergence of L N to A.
5 Proofs of Theorems 1.1, 1.2 and 1.5 Lemma 5.1 Let (X 1 , X 2 , A) be any limit point of (X 1,N , X 2,N , L 2,N ). Then the collision local time L(X 1 , X 2 ) exists and for any φ ∈ C b (R d ), where the stochastic integral term is a continuous local martingale with quadratic variation As in Section 5 of Barlow, Evans and Perkins (1991) Now apply Lemmas 4.7, 4.8, 4.9, 4.10, 4.11 (trivially dropping the non-negativity hypothesis on φ), and argue as in Section 5 of Barlow, Evans, Perkins (1991), essentially using dominated convergence, to show that all the terms (5.2) (except possibly the last one) converge in probability to the corresponding terms of (5.1), as ↓ 0. Hence the last term in (5.2), L t (X 1 , X 2 )(φ), converges in probability to say L t (φ) for each φ ∈ C b (R d ). This gives (5.1) with L t (φ) in place of L t (X 1 , X 2 )(φ). As each term in (5.1) is a.s. continuous in t (use (4.16) for the left-hand side) the same is true of t → L t (φ). This implies uniform convergence in probability for t ≤ T of L t (X 1 , X 2 ) to L t (φ) for each φ ∈ C b (R d ). It is now easy to construct L as a random non-decreasing continuous M Fvalued process, using a countable convergence determining class and hence we see that by definition L t = L t (X 1 , X 2 ).
Proof of Theorem 1.1 In view of Proposition 3.1, it only remains to show that L t (X 1 , X 2 ) exists and equals A t . This, however now follows from Lemma 5.1, Proposition 4.1, and the uniqueness of the decomposition of the continuous semimartingale X t (G α φ).
Remark 5.2 Since A t = L t (X 1 , X 2 ), Lemma 5.1 immediately gives us the following form of Tanaka's formula for (X 1 , X 2 ): where the stochastic integral term is a continuous local martingale with quadratic variation Proof of Theorem 1.2 As mentioned in the Introduction, the case of γ 2 ≤ 0 is proved in Theorem 4.9 of Evans and Perkins (1994) and so we assume γ 2 > 0. Let X i 0 satisfy UB, i = 1, 2, and assume P is a law on C M F 2 under which the canonical variables (X 1 , X 2 ) satisfy MP 0,γ 2 Then X 1 is a super-Brownian motion with branching rate 2, variance parameter σ 2 , and law P X 1 , say. By Theorem 4.7 of Barlow, Evans and Perkins (1991), Let q η : R + → R + be a C ∞ function with support in [0, η] such that q η (u)du = 1. For ε > 0 we will choose an appropriate η = η(ε) ≤ ε below and so may define Let E r,x denote expectation with respect to a Brownian motion B beginning at x at time r and let P r,t f (x) = E r,x (f (B t )) for r ≤ t. It is understood that B is independent of (X 1 , X 2 ). Define where the integrand is understood to be 0 for s < r under P r,x . The additional smoothing in time in our definition of h ε does force some minor changes, but it is easy enough to modify the arguments in Theorems 4.1 and 4.7 of Evans and Perkins (1994), using (5.4), to see there is a Borel : and for a.a. X 1 , t → t (B, X 1 ) is an increasing continuous additive functional of B. It is easy to use (5.4) and (5.5) to see that for a.a. X 1 , t is an admissible continuous additive functional in the sense of Dynkin and Kuznetsov (see (K 1 ) in Kuznetsov (1994)). Let φ as a finite limit at ∞}, and C + be the set of non-negative functions in C . Let φ : C M F → C + be Borel and let V r,t = V X 1 r,t φ be the unique continuous C + -valued solution of The existence and uniqueness of such solutions is implicit in Kuznetsov (1994) and may be shown by making minor modifications in classical fixed point arguments (e.g. in Theorems 6.1.2 and 6.1.4 of Pazy (1983)), using the L 2 bounds on t from (5.5) and the negative quadratic term in (5.6) to see that explosions cannot occur. The construction shows V X 1 r,t is Borel in X 1 -we will not comment on such measurability issues in what follows. Theorems 1 and 2 of Kuznetsov (1994) and the aforementioned a.s. admissibility of (B, X 1 ) give the existence of a unique right continuous measure-valued Markov process X such that (5.7) E X 0 (e −Xt(φ) ) = e −X 0 (V 0,t φ) , φ ∈ C + .
If P X 0 |X 1 is the associated law on path space we will show (5.8) P (X 2 ∈ ·|X 1 ) = P X 2 0 |X 1 (·) and hence establish uniqueness of solutions to MP 0,γ 2 X 1 0 ,X 2 0 . The proof of the following lemma is similar to its discrete counterpart, Proposition 3.5(b), and so is omitted ((5.4) is used in place of Proposition 2.4).
Let D(∆/2) be the domain of the generator of B acting on the Banach space C and D(∆/2) + be the nonnegative functions in this domain. Assume now that φ : C M F → D(∆/2) + is Borel. Let V ε r,t = V ε,X 1 r,t , be the unique continuous C + -valued solution of We claim V ε r,t also satisfies Theorem 6.1.5 of Pazy (1983) shows that V ε r,t ∈ D(∆/2) for r ≤ t, is continuously differentiable in r < t as a C -valued map, and satisfies Then (5.11) is just the mild form of (5.12) and follows as in section 4.2 of Pazy (1983). Note here that h ε is continuously differentiable in s thanks to the convolution with q η and so Theorem 6.1.5 of Pazy (1983) does apply. We next show that First use (5.11) and Lemma 5.3 to see that for each t > 0, Using (5.11) again, we see that (5.14) shows that → 0 uniformly in r ≤ t as ε, ε ↓ 0 P X 1 − a.s. (by (5.5) and Lemma 5.3).
(5.9) implies T ε,ε 1 → 0 uniformly in r ≤ t as ε, ε ↓ 0 and (5.14) also implies The above bounds and a simple Gronwall argument now show that V ε r,t (x) converges uniformly in x ∈ R d , and r ≤ t to a continuous C + -valued map as ε ↓ 0. It is now easy to let ε ↓ 0 in (5.10) and use (5.5) to see that this limit is V r,t , the unique solution to (5.6). This completes the derivation of (5.13).
As usual F X i t is the canonical right-continuous filtration generated by X i . Consider also the enlarged filtrationF t = F X 1 ∞ × F X 2 t . Argue as in the proof of Theorem 4.9 of Evans and Perkins (1994), using the predictable representation property of X 1 , to see that for φ ∈ D(∆/2), It is easy to extend the martingale problem for X 2 to bounded f :  (2002)). For such an f one has To see this, for f : Note that if f n → f in the bounded pointwise sense where the bound may depend on X 1 , then ∂f δ n ∂u → ∂f δ ∂u and ∆ 2 f δ n → ∆ 2 f δ in the same sense as n → ∞. By starting with f (u, x, X 1 ) = f 1 (u, x)f 2 (X 1 ) where f i are bounded and Borel, and using a monotone class argument we obtain (5.16) for f δ where f is any Borel map on R + × R d × C M F with sup s≤t,x |f (s, x, X 1 )| < ∞ for each X 1 . If f is as in (5.17) then it is easy to let δ ↓ 0 to obtain (5.16) for f . Note we are using the extension of the martingale measure M 2 to the larger filtrationF t in these arguments. Now recall φ : C M F → D(∆/2) + (Borel) and V ε r,t is the unique solution to (5.10). Recall from (5.12) that V ε r,t is a classical solution of the non-linear pde and in particular (5.17) is valid for f (r, x) = V ε r,t (x). Therefore we may use (5.12) in (5.16) to get We claim the last term in (5.18) approaches 0 uniformly in s ≤ t P -a.s. as ε = ε k ↓ 0 for an appropriate choice of η k = η(ε k ) in the definition of h ε . The definition of collision local time allows Let δ 0 > 0. By (5.13) and the uniform continuity of (r, x) → V r,t (x) there is a k 0 = k 0 (X 1 ) ∈ N a.s. such that sup r≤t |V r,t x 1 + x 2 2 − V ε k r,t (x 2 )| < δ 0 for k > k 0 and |x 1 − x 2 | < ε k 0 .
By considering |x 1 − x 2 | < ε k 0 and |x 1 − x 2 | ≥ ε k 0 separately one can easily show there is a k 1 = k 1 (X 1 ) so that sup Next use the upper bound in (5.14) and the continuity of u → p ε k * X 1 u (x 2 ) to choose η so that η k = η(ε k ) ↓ 0 fast enough so that sup s≤t I k 2 (s) → 0 P -a.s. as k → ∞.
The above bounds show the lefthand side of (5.19) converges to 0 a.s. as k → ∞. The a.s. convergence of L ε k (X 1 , X 2 ) to L(X 1 , X 2 ) therefore shows that The uniform convergence in (5.13) now shows that the last term in (5.18) approaches 0 uniformly in s ≤ t P -a.s. as ε = ε k → 0. (5.13) also allows us to let ε k ↓ 0 in (5.18), taking a further subsequence perhaps to handle the martingale term, and conclude Now apply Ito's lemma to conclude The stochastic integral is a boundedF t -local martingale and therefore is anF t -martingale. This proves for t 1 < t, This uniquely identifies the joint distribution of (X 2 t 1 , X 2 t ) conditional on X 1 . Iterating the above a finite number of times, we have identified the finite-dimensional distributions of X 2 conditional on X 1 , and in fact have shown that conditional on X 1 , X 2 has the law of the measure-valued process considered by Dynkin and Kuznetsov in (5.7).
Proof of Theorem 1.5. Let (X 1 , X 2 ) be a solution to M −γ 1 ,γ 2 X 1 0 ,X 2 0 and let P denote its law on the canonical space of M F 2 -valued paths. Conditionally on X, let Y denote a super-Brownian motion with immigration γ 1 L t (X 1 , X 2 ) (see Theorem 1.1 of Barlow, Evans and Perkins (1991)) constructed perhaps on a larger probability space. This means for φ ∈ C 2 b (R 2 ), and M Y is orthogonal with respect to the M X i , i = 1, 2. All these martingales are martingales with respect to a common filtration. Then it is easy to check thatX 1 = X 1 + Y satisfies the martingale problem characterizing super-Brownian motion starting at X 1 0 , i.e., is as in the first component in M 0,0 . Therefore there is jointly continuous function,ū 1 (t, x), with compact support such thatX 1 t (dx) =ū 1 (t, x)dx (see, e.g., Theorem III.4.2 and Corollary III.1.7 of Perkins (2002)) and so there is a bounded function on compact support, u 1 (t, x), so that X 1 t (dx) = u 1 (t, x)dx by the domination X 1 ≤X 1 . Let φ ∈ C b (R d ). Then Lebesgue's differentiation theorem implies that lim δ→0 p δ (x 1 − x 2 )φ( x 1 + x 2 2 )u 1 (s, x 1 )dx 1 = φ(x 2 )u 1 (s, x 2 ) for Lebesgue a.a. (s, x 2 ) a.s.
Moreover the approximating integrals are uniformly bounded by φ ∞ u 1 ∞ and so by Dominated Convergence one gets from the definition of L(X 1 , X 2 ) that Evans and Perkins (1994) (Theorem 3.9) used Dawson's Girsanov theorem to show there is a unique in law solution to M −γ 1 ,0 X 1 0 ,X 2 0 in our one-dimensional setting. If P −γ 1 ,0 denotes this unique law on the canonical path space of measures, then they also showed (5.20) P −γ 1 ,0 << P X 1 0 × P X 2 0 , the product measure of two super-Brownian motions with diffusion parameter σ 2 and branching rate 2. Our boundedness of u 1 shows that The latter is a special case of our argument when γ 2 = 0. This allows us to apply Dawson's Girsanov theorem (see Theorem IV. 1.6 (a) of Perkins (2002)) to conclude that Here M X 2 is the martingale measure associated with X 2 and u 1 is the density of X 1 , both under P −γ 1 ,0 . Although the Girsanov theorem quoted above considered absolute continuity with respect to P X 1 0 × P X 2 0 , the same proof gives the above result. This proves uniqueness of P and, together with (5.20) shows that P is absolutely continuous with respect to P X 1 0 × P X 2 0 . This gives the required properties of the densities of X i as they are well-known for super-Brownian motion (see Theorem III.4.2 of Perkins (2002)).

Proof of the Concentration Inequality-Proposition 2.4
As we will be proving the concentration inequality for the ordinary rescaled critical branching random walk,X 1,N , in order to simplify the notation we will write X N forX 1,N , and write ξ N , or just ξ, forξ 1,N . Dependence of the expectation on the initial measure X N 0 =X 1,N 0 will be denoted by E X N 0 . {P N u , u ≥ 0} continues to denote the semigroup of our rescaled continuous time random walk B N .
Notation. If ψ : Z N → R, let P N ψ(x) = y p N (y − x)ψ(y) and let To bound the mass in a fixed small ball we will need good exponential bounds. Here is a general exponential bound whose proof is patterned after an analogous bound for super-Brownian motion (see e.g. Lemma III.3.6 of Perkins (2002)). The discrete setting does complicate the proof a bit. Proposition 6.1 Let f : Z N → R + be bounded and definē If t > 0 satisfies A short calculation using Ito's lemma for Poisson point processes (see p. 66 in Ikeda and Watanabe (1981)) shows that andM N is a locally bounded local martingale. In fact Assume now that for some c 0 > 0, and note that if |w| ≤ 4c 0 , then Now assume t > 0 satisfies (6.1), let c 1 = 7 exp (4 f ∞ /N ) and define κ( As convoluting P N with P N t amounts to running B N until the first jump after time t, one readily sees that these operators commute and hence so do R and P N t . Therefore We also have . (6.9) By (6.1) and (6.8), for u ≤ t, and so (6.5) holds with c 0 = f ∞ N . Clearly φ,φ ∈ C b ([0, t] × Z N ) and so (6.7) is valid with this choice of c 0 . We therefore have Rf ( by (6.9)) ≤ 0, the last by the definition of c 1 . Now return to (6.4) with the above choice of φ. By choosing stopping times T N k ↑ ∞ as k → ∞ such that E(M N t∧T N k ) = 0 and using Fatou's lemma we get from the above that (by (6.1)).
We now specialize the above to obtain exponential bounds on the mass in a ball of radius r. In fact we will use this bound for the ball in a torus and so present the result in a more general framework. Lemma 7.3 below will show that the key hypothesis, (6.10) below, is satisfied in this context.
Proof Take f ≡ θ (θ as above) in Proposition 6.1. Note that condition (6.1) holds iff t ≤ exp(−4θ/N )(14θ) −1 and so for θ ≤ N is implied by our bound on t. As P N t Rf = 2θ, Proposition 6.1 gives the result.
Remark 6.4 The above corollary is of course well known as X N t (1) = Z N t /N , where {Z u , u ≥ 0} is a rate 1 continuous time Galton-Watson branching process starting with N X N 0 (1) particles and undergoing critical binary branching. It is easy to show, e.g. by deriving a simple non-linear o.d.e in t for E(u Zt |Z 0 = 1), (see (9.1) in Ch. V of Harris (1963)) that for θ > 0 and t < [N (e θ/N − 1)] −1 , or θ ≤ 0 and all t ≥ 0, .
The above exponential bounds will allow us to easily obtain the required concentration inequality on a mesh of times and spatial locations. To interpolate between these space-time points we will need a uniform modulus of continuity for the individuals making up our branching random walk. For this it will be convenient to explicitly label these individuals by multi-indices If β = (β 0 , . . . , β n ) ∈ I, let πβ = (β 0 , . . . , β n−1 ) if n ≥ 1 be the parent index of β, set πβ = ∅ if n = 0, and let β|i = (β 0 , . . . , β i ) if 0 ≤ i ≤ n ≡ |β|. Define β|(−1) = ∅. Let {τ β : β ∈ I}, {b β : β ∈ I} be two independent collections of i.i.d. random variables with τ β exponentially distributed with rate 2N and P (b β = 0) = P (b β = 2) = 1/2. Let T ∅ = 0 and define We will think of [T πβ , T β ) as the lifetime of particle β, so that T β = T πβ (iff b β|j = 0 for some j < |β|) means particle β never existed, while b β is the number of offspring of particle β. Write β ∼ t iff particle β is alive at time t, i.e., iff be a collection of independent copies of B N , starting at 0. Formally they may be defined inductively with B β T πβ chosen to be B πβ T πβ − . Such labelling schemes may be found in Ch. 8 of Walsh (1986) or Ch. II of Perkins (2002).
Ifξ(x) = β∼t 1(B β t = x), then (ξ t , t ≥ 0) and (ξ t , t ≥ 0) are identical in law. One can see this by noting {ξ t : t ≥ 0} is an S F -valued Markov process with the same jump rates and initial condition as {ξ t : t ≥ 0}. Formally one can work with the associated empirical processesX N t = 1 N xξ t (x) and X N t and calculate their respective generators as in Section 9.4 of Ethier and Kurtz (1986). The generator of the former can be found by arguing as in Section II.4 of Perkins (2002).
Alternatively, one can in fact define the above branching particle system from our original Poisson equations (2.2) since one can use the uniform variables in our driving Poisson point processes to trace back ancestries. We briefly outline the construction. We begin by labeling the |ξ 0 | initial particles as above with multi-indices 0, . . . , |ξ 0 | − 1 ∈ I and assigning each particle at each site x an integer 1 ≤ k ≤ ξ 0 (x) that we call its level. Since there are only finitely many particles we can explain how these labels and levels propagate forwards in time at the jump times in (2.2). If at time t there is an "arrival" in Λ 1,+ x with height u ∈ [k − 1, k), where k ≤ ξ t− (x), then the particle at level k at site x branches. If β is the label of this branching particle at time t− then two new particles are created at x with labels β0 at level k and β1 at level ξ t (x) = ξ t− (x) + 1. (β no longer labels a particle at time t.) All other particles keep their current labels and levels. If at time t there is an "death" in Λ 1,− x with height u ∈ [k − 1, k), where k ≤ ξ t− (x), then the particle at level k at site x dies. It is removed from the population (it's label β no longer labels a particle at time t), and all particles at x with levels greater than k have their levels decreased by 1, and keep their current labels at time t. If at time t there is a "migration" from y to x in Λ 1,m x,y with height u ∈ [k − 1, k), where k ≤ ξ t− (y), then the particle at level k at site y migrates to x where its new level is ξ t (x) = ξ t− (x) + 1. The particles at y with levels greater than k have their levels reduced by 1. The migrating particle keeps its label β, so that this label now refers to particle ξ t (x) at site x. All other particles keep their current labels and levels. At this point we have inductively defined a multi-index α(t, x, k) ∈ I which labels the particle with level k ≤ ξ t (x) at site x ∈ Z N and time t ≥ 0. Here we use α to denote this random function as β will denote the independent I-valued variable. As the birth of a label coincides with a branching event and its death must coincide with either a branching event (it has 2 children) or a death event (it has 0 children) it is clear the set of times at which β is alive, A β = {t ≥ 0 : ∃x, k s.t. α(t, x, k) = β} is a left semi-closed interval [U β , T β ) (possibly empty) with U β = T πβ , where again T ∅ = 0. Also we see that T β −T πβ is exponential with rate 2N (the time to a birth or death event of the appropriate height corresponding to the level labelled by β). The independence properties of the Poisson point processes allow one to show that the collection of these exponential increments are independent. They are only indexed by those β which have a positive lifespan but the collection can be padded out with iid exponentials to reconstruct the indendent τ β 's described above. The same reasoning applies to reconstruct the b β 's. For t ∈ [T πβ , T β ) define B β t and β t by Then B β starts at B πβ T πβ − and jumps according to migration events of the appropriate height. Therefore conditional on the {T β , b β } it is a copy of B N . The independence property of the Poisson point processes Λ 1,m x,y show they are (conditionally on the birth and death times) independent. We have shown that the collection of labelled branching particles constructed from (2.2) are identical in law to those described above and used for example in Ch. II of Perkins (2002). We have been a bit terse here as the precise details are a bit tedious and in fact one can proceed without this intrinsic labelling as noted above since starting from the labelled system one can reconstruct the process of interest, X N .
However you prefer to proceed, it will be convenient to extend the definition of B β s to all s ∈ [0, T β ) by following the paths of β's ancestors, i.e., Then for β ∼ t, s → B β s is a copy of B N on [0, T β ) as it is a concatenation of independent random walks with matching endpoints.
Here is the modulus of continuity we will need.
Proof In Section 4 of Dawson, Iscoe and Perkins (1989) a more precise result is proved for a system of branching Brownian motions in discrete time. Our approach is the same and so we will only point out the changes needed in the argument. Let h(r) = r 1 2 −ε . In place of the Brownian tail estimates we use the following exponential bound for our continuous time random walks which may be proved using classical methods: (6.16) Π N 0 (|B N (2 −n )| > h(2 −n )) ≤ 3d exp −c 6.16 2 nε for 2 −n ≥ N −1 and some positive c 6.16 (ε).
Set δ N,ε = 2 −n iff ω / ∈ A * n 1 ∪ (∪ n 1 n =n A n ) and ω ∈ A n−1 for n ∈ N. If ω ∈ A * n 1 , set δ N,ε = 1. A standard binomial expansion argument (as in Dawson, Iscoe and Perkins (1989)) shows that (6.15) holds with some universal constant c in front of the bound on the righthand side. The latter is easily removed by adjusting ε. It follows easily from (6.16) and (6.17) that P (A n ) ≤ c 1 exp −c 2 2 nε . Therefore to show (6.14), and so complete the proof it remains to show This bound is the only novel feature of this argument. It arises due to our continuous time setting. Let 0 ≤ j ≤ n 1 2 n 1 and condition on N j = N X N j2 −n 1 (1). For each of these N j particles run it until its first branching event. The particle is allowed to give birth iff it splits into two before time (j + 1)2 −n 1 . If it splits after this time or if it dies at the branching event, it is killed. Note that particles will split into two with probability the last inequality by our definition of n 1 . Let Z 1 be the size of the population after this one generation. Now repeat the above branching mechanism allowing a split only if it occurs before the next time interval of length 2 −n 1 and continue this branching mechanism until the population becomes extinct. In this way we get a subcritical discrete time Galton-Watson branching process with mean offspring size µ = 2q < 1. Let T j denote the extinction time of this process. Then P (T j > m) ≤ E(Z m ) = N j µ m and so, integrating out the conditioning, we have (6.19) P ( max In keeping track of this branching process until T j we may in fact be tracing ancestral lineages beyond the interval [j2 −n 1 , (j + 1)2 −n 1 ], which is of primary interest to us, but any branching events in this interval will certainly be included in our Galton Watson branching process as they must occur within time 2 −n 1 of their parent's branching time. The resulting key observation is therefore that T j is an upper bound for the largest number of binary splits in [j2 −n 1 , (j + 1)2 −n 1 ] over any line of descent of the entire population. Since the offspring can move at most 1/ √ N from the location of its parent at each branching event, T j / √ N gives a (crude) upper bound on the maximum displacement of any particle in the population over the above time interval. It therefore follows from (6.19) that P max Now take m to be the integer part of √ N h(2 −n 1 ) in the above. A simple calculation (recall Assumption 2.5) now gives (6.18) and so completes the proof.
Proof of Proposition 2.4. Let ε ∈ (0, 1/2) and T ∈ N. We use n 0 = n 0 (ε, T ) to denote a lower bound on n which may increase in the course of the proof. Let h(r) = r 1 2 −ε (as above) and φ(r) = r 2∧d−ε for r > 0. Following Barlow, Evans and Perkins (1991), let B n (y) = B(y, 3h(2 −n )), B n = {B n (y) : y ∈ 2 −n Z d } and for n ≥ n 0 define a class C n of subsets of R d such that from some constants c 6.22 and c 6.23 , depending only on {X N 0 } and ε, To construct C n note first that as in Lemma 4.2 for B ∈ B n , Here we have used Assumption 2.5. For a fixed y ∈ (2 −n Z d ) ∩ [0, 1) d let C be a union of balls in B n of the form k + B n (y), k ∈ Z d , where such balls are added until max 0≤j≤T 2 n µ N j2 −n (C) > φ(h(2 −n )). It follows from (6.24) that for each 0 ≤ j ≤ 2 n T , µ j2 −n (C) ≤ φ(h(2 −n )) + c 0 φ(h(2 −n )) and so (6.22) holds with c 6.22 = 1 + c 0 . Continue the above process with new integer translates of B n (y) until every such translate is contained in a unique C in C n . If n 0 is chosen so that 6h(2 −n 0 ) < 1, then these C's will be disjoint and all but one will satisfy max 0≤j≤2 n T µ j2 −n (C) > φ(h(2 −n )). Therefore number of C's constructed from B n (y) ≤ 2 n T j=0 µ j2 −n (B n (y))φ(h(2 −n )) −1 + 1 (6.25) ≤ X N 0 (1)T 2 n φ(h(2 −n )) −1 + 1. Now repeat the above for each y ∈ (2 −n Z d ) ∩ [0, 1) d . Then (6.25) gives (6.23), and (6.20) and (6.21) are clear from the construction.
Choose 0 ≤ j ≤ 2 n T and y ∈ 2 −n Z d so that t ∈ [j2 −n , (j + 1)2 −n ) and |y − x| ≤ 2 −n ≤ h(2 −n ), respectively. Let β ∼ t and B β t ∈ B(x, h(2 −n ). Then Proposition 6.5 implies and so, where the last inclusion holds by (6.20). For C ∈ C n , let denote the time t distribution of descendents of particles which were in C at time j2 −n . Use (6.26) to see that X N t (B(x, h(2 −n ))) ≤ X N,j2 −n ,C t (1), and therefore (6.27) sup The process t → X N,j2 −n ,C t+j2 −n (1) evolves like a continuous time critical Galton-Watson branching process-conditional on F j2 −n it has law P X N j2 −n | C -and in particular is a martingale. If θ n , λ n > 0 we may therefore apply the weak L 1 inequality to the submartingale exp(θ n X N,j2 −n ,C t+j2 −n (1)) to see that (6.27) implies (6.28) P sup . For n ≥ n 0 we may assume 2 −n ≤ e −4 (14θ n ) −1 , and as we clearly have θ n ≤ 2 n ≤ N , Corollary 6.3 implies (6.29) E X N j2 −n | C (exp (θ n X N 2 −n (1))) ≤ exp(4θ n X N j2 −n (C)).
Use this in (6.29) and apply the resulting inequality and (6.23) in (6.28) to see that for n ≥ n 0 (ε, T ), ≤ (2 n T + 1) c 6.23 T exp(c(d)n) exp (−2 nε )e 2c 6.22 + P (δ N ≤ 2 −n ) We have used (6.14) from Proposition 6.5 in the last line. If r 0 (ε, T ) = h(2 −n 0 ), the above bound is valid for (recall n ≥ n 0 and 2 −n ≥ N −1 ) A calculation shows that 2 −n(1−4ε) ≤ h(2 −n ) d∧2−8ε 2 −nε . Therefore elementary Borel-Cantelli and interpolation arguments (in r) show that if then for any η > 0 there is an M = M (η, , T ) such that Here we are also using by the weak L 1 inequality. The latter also allows us to replace r 0 (ε) by 1 in the above bound. If we set r = r that is H 10ε,N (0, T, 1) ≤ H 8ε,N (ε, T, 1). Therefore we have shown that for each T ∈ N and ε > 0, H 10ε,N (0, T, 1) = sup 0≤t≤T ρ N 10ε (X N t ) is bounded in probability uniformly in N . The fact that X N · has a finite lifetime which is bounded in probability uniformly in N (e.g. let θ → −∞ in (6.13)) now lets us take T = ∞ and so complete the proof of Proposition 2.4. 7 Proof of Proposition 3.5 and Lemma 3.7 In this section we will verify some useful moment bounds on the positive colicin models which, through the domination in Proposition 2.2, will be important for the proof of our theorems. As usual, |x| is the L ∞ norm of x.
Lemma 7.1 Let Then there exist c, C > 0 such that Proof It suffices to bound P (S n / √ n = x) for x ∈ Z d /M √ n by the right-hand side with r = n −1/2 for then we can add up the points in the cube to get the result. For this, note if |x| ≤ 2 we can ignore the exponentials on the right-hand side, and for |x| > 2 we have |x | > |x|/2 for all x ∈ x + [−r, r] d and any r ≤ 1. The local central limit theorem implies Suppose now that n = 2m is even. For 1 ≤ i ≤ d, (7.3) P (|S i n | ≥ an, S n = y) ≤ 2P (|S i m | ≥ am, S n = y) since for |S i n | ≥ an we must have |S i m | ≥ am or |S i n −S i m | ≥ am and the increments of S i k conditional on S n = y are exchangeable. A standard large deviations result (see Section 1.9 in Durrett (2004) Let y = x √ n and choose i so that |y i | = |y|. Taking a = |x|/ √ n it follows that so using (7.2) and (7.3) we have The result for odd times n = 2m + 1 follows easily from the observation that if S n = x then S n−1 = y where y is a neighbor of x and on the step n the walk must jump from y to x.
Using the local central limit theorem for the continuous time random walk to estimate the probability of being at a given point leads easily to: Lemma 7.2 There is a constant c such that for all s > 0 (7.4) and in particular for r = 1/ √ N and any N ≥ 1 we have We also used the following version of Lemma 7.2 for the torus in the proof of the concentration inequality in the previous section. This time we give the details.
If {S n } is as in Lemma 7.1, we may define B N u = S τ N u / √ N , where τ N is an independent rate N Poisson process. We have Standard exponential bounds imply for some c > 0. By (7.6), we have j −1/2 ≤ r N j ≤ 1 for N u/2 ≤ j ≤ 2N u and so we may use Lemma 7.1 to bound the series on the right-hand side of (7.7) by In the last line we have carried out an elementary calculation (recall |x| ≤ 1) and as usual are changing constants from line to line. Use the above and (7.8) in (7.7) to get since r ≥ N −1/2 . The result follows.
Eliminating times near 0 and small |x| we can improve Lemma 7.2.
Proof As before, it suffices to consider r = N −1/2 . Let τ N s = number of jumps of B N up to time s. Then, as in 7.8, there exists c > 0 such that (7.9) Π N 0 τ N s > 2N s or τ N s < N s/2 ≤ e −csN . Now for N s/2 ≤ j ≤ 2N s and r ≥ N −1/2 , Lemma 7.1 implies Using 2N s ≥ j ≥ N s/2, |x| ≥ s 1/2−δ * , and calculus the above is at most To combine this with (7.9) to get the result, we note that (s a ∧ 1) Using e −cN δ /2 ≤ c(δ )N −d/2 we see that the right-hand side of (7.9) is also bounded by and the proof is completed by combining the above bounds.
Lemma 7.6 There exists c 7.11 = c 7.11 (δ,δ) such that for all s > 0, z ∈ Z N and µ N ∈ M F (Z N ) Proof Recall l d = (d/2 − 1) + . Let s ≥ 1. Then from Lemma 7.2 we immediately get that and hence for s ≥ 1 we are done. Now let 0 < s < 1 and fix 0 < δ * < 1/2 such that δ/2 + δ * ((2 ∧ d) − δ) =δ. A little algebra converts the condition into (7.12) Define A N = y : |z − y| ≤ (s 1/2−δ * ∨ N −1/2+δ * ) + N −1/2 . Let I 1 N and I 2 N be the contributions to (7.11) from the integrals over A N and A c N , respectively. Let s ≥ 1/N . Then we apply Lemma 7.2 to bound I 1 N as follows: where (7.12) is used in the last line. If s ≤ 1/N we get (using (7.12) again) As for the second term I 2 N we have the following. Note that for any y ∈ A c N and |z 1 | ≤ 1/ √ N we have, Then by (7.13) and Lemma 7.5 we get where the last inequality is trivial since s < 1. Summing our bounds on I 1 N and I 2 N gives the required inequality for 0 < s < 1 and the proof is complete.
Return now the the proof of Proposition 3.5. Let φ : Z → [0, ∞). By our choice ofδ, From (2.19), Lemma 7.6, and the definition of H δ,N we get the following bounds by using the Markov property of B N : and for µ N ∈ M F (Z N ), In the above integrals, s 0 = 0 and we have distributed theX 1,N s i integrations among the B N increments in different manners in (7.14) and (7.15). Define

Some elementary calculations give
and we are done.
Proof of Lemma 3.7 (a) Recall that our choice of parameterδ in (3.5) implies thatl d = 1 − l d −δ > 0. By Proposition 3.5(a) we get By Lemma 7.6 the first term in (7.21) is bounded by uniformly on x 1 . Now it is easy to check that t 0 s −2δ (t − s) −l d −δ ds ≤ c(t), and c(t) is bounded uniformly on the compacts. (7.23) Apply (7.23) and Lemma 7.6, and recall thatR N (t) may change from line to line, to show that the second term in (7.21) is bounded by Now put together (7.22, (7.24) to get that (7.21) is bounded bŷ and we are done.
(b) Apply Lemma 7.7 with φ(x) = N d/2 By Lemma 7.6 the first term in (7.25) is bounded by uniformly on x 1 . Let us consider the second term in (7.25). First apply Lemma 7.6 to bound the integrand.
Hence the second term in (7.25) is bounded bŷ Now apply (7.23) and Lemma 7.6 to show that the third term in (7.25) is bounded by Now put together (7.26), (7.27), (7.28) to bound (7.25) bŷ By Lemma 7.2 we get the following bound on I 1,N : As for I 2,N , apply Lemma 7.6 to get By combining (8.1) and (8.2) we are done.
Lemma 8.1 There is a c = c α > 0 such that Proof First apply Lemma 7.2 to get .

Now a trivial calculation shows that
and we are done.
Notation, assumptions. Until the end of this section we will make the following assumption on N, We will typically reserve notation µ N for measures in M F (Z N ) and c(t) will denote a constant depending on t ≥ 0 which is bounded on compacts. Also recall that δ,δ satisfy (3.5).
Proof First let us treat the case t ≤ η . By Lemma 8.1, for all x 1 , x 2 ∈ Z N , Now let us turn to the case t > η . Then Then for any x 1 where the last inequality follows by the definition of h d . Now apply again Lemma 7.2 and the assumption t > η to get and we are done.
We will also use the trivial estimate The proof of the following trivial lemma is omitted.
Then for 0 < η < 1/2 and some c(η, t), bounded for t in compacts, we have The next several lemmas will give some bounds on the Green's functions G α N used in Section 4.
Proof First, by Lemma 7.7 we get Note that I 1,N (s, t) = I 1,N (t − s), and hence, if t − s ≥ η , then by Lemma 8.2, I 1,N (s, t) in (8.7) is bounded by where the inequality follows since 1 − dη/2 ≥ η(1 − l d − 3δ) by our assumptions on η,δ. To bound I 1,N (s, t) for t − s ≤ η use again Lemma 8.2 and hence obtain Next consider I 2,N (s, t). First bound the integrand: (8.10) where the last inequality follows by Lemma 7.6 and definition of R N . Now for t − s ≤ η we apply (8.10), (8.5), and Lemma 8.2 to get Now, for t − s ≥ η use (8.10) to get Let us estimate the first integral on the right hand side of (8.12). By Lemma 8.2 with 2η < 2/3 in place of η, it is bounded by The last line follows as in (8.8) and uses η < 2/7. Now, let us estimate the second integral on the right hand side of (8.12). Since t − s ≥ η , we see that for < 1/2, there is a constant c = c(η) such that t − 2η ≥ s + η − 2η ≥ s + c η , and hence the second integral on the right hand side of (8.12) is bounded by x ψ N (x 2 , B N t−s 1 ) ds 1 (8.14) ≤ c η(−l d −δ) 2η(1−l d −δ) (by (8.5) and Lemma 8.2).
Lemma 8.5 There is a c = c α such that Proof By Lemma 7.2 we get Lemma 8.6 There is anR N (t) so that for all t ≥ s ≥ 0, x 1 ∈ Z N , µ N ∈ M F (Z N ), and N ∈ N .
Proof The result is immediate by Lemma 8.5 and Proposition 3.5(b).
Lemma 8.7 There is a c 8.7 such that if µ N ∈ M F (Z N ), then N δ (µ N ) , ∀t ≥ 0, x ∈ Z N , N ∈ N.
Proof Use (8.20) to get (8.21) for all t ≥ 0, x, x 1 , z 1 ∈ Z N . If µ N ∈ M F (Z N ), we have (8.22) where the last inequality follows by Lemma 7.6. This and (8.21) imply and the proof is finished.
Lemma 8.8 There is anR N (t) such that for all t ≥ 0, x ∈ Z N , µ N ∈ M F (Z N ), N ∈ N .
We next need a bound on the convolution of P g N s,t with p N . Let g N (X 1,N s , x) = 2 −d N d/2X 1,N s (B(x, 2N −1/2 )).
The first term on the right hand side of (8.30) is bounded by Z 2 N P g N 0,t (ψ N (·, x)) (x 1 )X 2,N 0 (dx 1 )X 1,N t (dx) (8.32) where the first inequality follows by Lemmas 8.9 and 3.7(a), and the second is immediate consequence of the definition ofˆ N δ (X 2,N 0 ). Now we consider the second term on the right-hand side of (8.30). By Lemma 8.4, for η ∈ (0, 2/7), it is bounded by where the last inequality follows by Lemma 3.7(a). This and (8.5) allows us to bound (8.33) by Now we consider the third term on the right-hand side of (8.30). We use Lemmas 8.4, 8.10 to show that it is bounded bȳ where in the last inequality we applied Lemma 3.7(a) for the Pḡ N semigroup. This, (8.5) and (8.36) imply that the third term is bounded bȳ Proof of Lemma 4.5 It is easy to use Lemma 2.3 and (3.7), as in (8.30), to show that N P gn 0,t (ψ N (x , ·)) (x 1 )P gn 0,t (G α N 1(x , ·)) (x 2 )X 2,N 0 (dx 1 )X 2,N 0 (dx 2 ) X 1,N t * q N (dx) + c 1 + H δ,N N −1+ d +δ/2 E t 0 sup z,x∈Z N P gn s,t (ψ N (x , ·)) (z) × Z N sup z∈Z N Z N P gn s,t (G α N 1(x , ·)) (z) X 1,N t * q N (dx) X 2,N s (dz 1 ) ds|X 1,N ≡ I 1,N + I 2,N . (8.38) In the above the sup over z in the last integrand avoids the additional convolution with p N we had in (8.30). First, apply Lemmas 8.8, 8.9, and 4.2 to get Now, let us bound I 2,N . By Lemmas 8.4, 8.6, and 4.2 we get for η ∈ (0, 2/7), Apply Corollary 3.6(b) to get (8.40) where (8.5) is used in the last line. Combine this with (8.39) to finish the proof.