A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate

Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian traveling wave.


Introduction
intro_sec An important problem in evolutionary biology is to understand how the fitness of individuals in a population increases over time as a result of beneficial mutations. Results in the biology literature indicate that in large populations, if individuals acquire beneficial mutations at a constant rate, then the overall fitness level of the population increases at a constant rate, known as the rate of adaptation, while the empirical distribution of the fitness levels of individuals in the population becomes approximately Gaussian. That is, the empirical distribution of the fitness levels of individuals in the population evolves over time like a Gaussian traveling wave. The idea of modeling the fitness distribution by a traveling wave goes back at least to the work of Tsimring,Levine,and Kessler [33]. Later works discussing the Gaussian shape for the traveling wave include [4,12,15,27,28,29,30].
Although the idea that the fitness distribution evolves as a Gaussian traveling wave is well established in the biology literature, the mathematically rigorous work on this problem has been considerably more limited. The main aim of this paper is to provide a first rigorous analysis in which a non-degenerate Gaussian traveling wave is observed in this context. Before we go into the details of our own results, we give a brief overview of the existing mathematical literature. A standard mathematical model involves a population of fixed size N in which each individual independently acquires beneficial mutations at the constant rate µ. Beneficial mutations increase an individual's fitness by s, so that an individual that has acquired k beneficial mutations, which we call a type k individual, has fitness max{0, 1 + s(k − m(t))}, where m(t) is the mean number of mutations of the individuals in the population at time t. Each individual independently dies at rate one, and when an individual dies, the parent of the new individual that is born is chosen at random from the population with probability proportional to fitness. A number of authors have studied models very similar to this one. Yu,Etheridge,and Cuthbertson [35] and Kelly [24] obtained rigorous results concerning the rate of adaptation for a very similar model, but did not establish a Gaussian shape for the fitness distribution. Durrett and Mayberry [17] considered, for a closely related model, the case in which s is constant and the mutation rate is N −α , where 0 < α < 1. They rigorously established traveling wave behavior. However, they considered mutation rates that are small enough that the number of distinct types present in the population at a typical time is a constant that does not tend to infinity with N , which means the traveling wave does not have a Gaussian shape. Schweinsberg [32] considered slightly faster mutation rates, so that the mutation rate tends to zero more slowly than any power of N . This work essentially made rigorous the heuristics developed by Desai and Fisher [15]. For the range of parameter values considered in [32], the traveling wave exhibits Gaussian-like tail behavior, in the sense that the logarithm of the ratio of the number of individuals with ℓ more mutations than average to the number of individuals with an average number of mutations is proportional to −ℓ 2 . However, at a typical time, most individuals have the same number of mutations, which means that the empirical distribution of the fitnesses of individuals in the population is actually converging to a point mass, rather than to a Gaussian distribution. Up to this point, as far as we know, the empirical distribution of the fitnesses of individuals in the population in this model has not been rigorously shown to converge to a Gaussian distribution for any range of values of the parameters µ and s.
For the fitness distribution to be approximately Gaussian, the mutation rate needs to be large enough that one type does not dominate the population at a typical time, as it does for the parameter values considered in [17,32]. We therefore consider a scenario in which the rate of beneficial mutations is large, but the additional selective benefit resulting from each mutation is small. This is related to the so-called infinitesimal model in quantitative genetics. See [2] for a recent mathematical treatment of the infinitesimal model and an extensive survey of the relevant biology literature. Rather than modeling the effects of individual mutations, we will model the fitness level of an individual over time as a Brownian motion. Also, to simplify the analysis, we will allow the offspring of individuals to evolve independently, rather than requiring a fixed population size.

The Model
The above considerations lead us to consider the following branching Brownian motion process, which is the model that we will study throughout the rest of the paper. Because we aim to prove a limit theorem, we will consider a sequence of processes indexed by n. We begin with some configuration of particles at time zero, which may depend on n. Each particle independently moves according to one-dimensional Brownian motion with drift −ρ n , where ρ n > 0. Also, any particle at the location x independently dies at rate d n (x), and splits into two particles at rate b n (x), where b n (x) − d n (x) = β n x (1.1) bndn for some β n > 0. In particular, note that the birth and death rates are the same for particles at the origin. This model is very similar to that of Neher and Hallatschek [27], and the main results of this paper are mathematically rigorous versions of some of the results in [27].
As indicated above, we view this process as modeling a population undergoing selection. With this interpretation, particles represent individuals in a population, and the position of the particle corresponds to the fitness level of the individual.

Main Results
Before stating our main results, we will need to introduce some assumptions and some notation. Given two sequences of positive real numbers (a n ) ∞ n=1 and (b n ) ∞ n=1 , we will write a n b n if a n /b n is bounded above by a positive constant, and a n ≪ b n if lim n→∞ a n /b n = 0. We will use the symbols and ≫ likewise. We also write a n ≍ b n if a n /b n is bounded both above and below by positive constants. We write that a n is O(1) if the sequence (a n ) ∞ n=1 is bounded and o(1) if lim n→∞ a n = 0. We will also use O(1) and o(1) for random sequences that are uniformly bounded above by deterministic sequences that are O(1) and o(1) respectively.
We will make the crucial assumption that lim n→∞ ρ 3 n β n = ∞.
(1.4) A3 The assumptions (1.2), (1.3), and (1.4) will be in effect throughout the rest of the paper, even when they are not explicitly mentioned. Note that (1.1) and (1.4) are satisfied, for example, if d n (x) = 1 and b n (x) = 1 + β n x for all x ≥ −1/β n , while d n (x) = −β n (x) and b n (x) = 0 for x < −1/β n . Beyond Section 1, to lighten notation, we will drop the subscripts and write ρ and β in place of ρ n and β n . However, it is important for the reader to keep in mind that these parameters do depend on n.
We will also need to consider the Airy function Ai(x) = 1 π ∞ 0 cos y 3 3 + xy dy.
The Airy function Ai has an infinite sequence of zeros (γ k ) ∞ k=1 which satisfy · · · < γ 2 < γ 1 < 0. The Airy function and particularly the quantity γ 1 will play an important role in what follows. It is known (see table 9.9.1 in [16]) that to three decimal places, γ 1 ≈ −2.338.
(1.5) zerovalue We will let N n (t) denote the total number of particles in the system at time t, and we will let X 1,n (t) ≥ X 2,n (t) ≥ · · · ≥ X Nn(t),n (t) denote the locations of the particles at time t. We imagine our system drawn with time on the vertical axis, so that the maximal particle is the right-most. Let L n = ρ 2 n 2β n − (2β n ) −1/3 γ 1 .
Our particles will generally be to the left of L n , and we call the area near L n the right edge. Note that since γ 1 < 0, L n is to the right of ρ 2 n /2β n . Define e ρnX i,n (t) and Z n (t) = e ρnX i,n (t) Ai((2β n ) 1/3 (L n − X i,n (t)) + γ 1 )1 {X i,n (t)<Ln} .
While the form of Z n (t) may seem mysterious at this point, this turns out to be a natural measure of the "size" of the process at time t. It turns out that, if we modify the process by killing particles that reach L n , then (Z n (t), t ≥ 0) is a martingale. In addition to the assumptions (1.2), (1.3), and (1.4) on the parameters, we will make two assumptions on the initial configuration of particles at time zero. We will assume that for all ε > 0, there is a δ > 0 such that for sufficiently large n, In other words, the sequence of random variables (ρ 3 n β −1/3 n e −ρnLn Z n (0)) ∞ n=1 , and its reciprocal sequence, is tight.
We will also assume that ρ 2 n e −ρnLn Y n (0) → p 0, (1.7) Yasm where → p denotes convergence in probability as n → ∞. Roughly speaking, the condition (1.7) ensures that the contribution to Z n (t) for small values of t will not be dominated by the descendants of a single particle at time zero, nor will it be dominated by particles that are far from L n at time zero. The two conditions (1.6) and (1.7) cannot be satisfied by a single particle at any location. However, they are satisfied, for example, by letting (u n ) ∞ n=1 be any sequence satisfying ρ −1 n ≪ u n ≤ β −1/3 n , and starting with e ρnun /u n ρ 3 n particles all located at L n − u n . We are now ready to state our main result, establishing the Gaussian shape for the distribution of particles on a suitable time scale. Here and throughout the paper, δ y denotes the unit mass at y, and ⇒ denotes convergence in distribution for random elements of the Polish space of probability measures on R endowed with the weak topology. (1. 8
This result shows that the empirical distribution of particles shortly after time ρ n /β n is approximately Gaussian, and stays that way at least for times of the order ρ n /β n . Note that this Gaussian distribution has mean zero. However, if we considered a translation of the model in which particles move according to branching Brownian motion with no drift, with time-dependent birth and death rates satisfying b(x, t) − d(x, t) = β n (x − ρ n t), then the Gaussian particle distribution at time t n would be centered at ρ n t n , giving rise to the Gaussian traveling wave behavior discussed above. Therefore, the drift parameter ρ n can be interpreted in this model as the speed at which the traveling wave advances. Because the fitness of the population, as measured by the branching rate, increases by β n whenever the traveling wave advances by one unit, the fitness of the population increases over time at rate v n = β n ρ n . Note that the variance of the fitness distribution is then σ 2 n = β 2 n (ρ n /β n ) = v n , in agreement with Fisher's Fundamental Theorem of Natural Selection [18].
Because, as we will see, some particles with unusually high fitness account for nearly all of the offspring that are still alive at a time ρ n /β n in the future, it is also of interest to understand the empirical distribution of particles close to the right edge. To do this, we consider an empirical measure in which a particle at x is weighted by e ρnx . This leads to the following result. (1.10) tcond For t > 0, define the random probability measure ξ n (t) = 1 Y n (t) e ρnX i,n (t) δ (2βn) 1/3 (Ln−X i,n (t) ) (1.11) xidef if N n (t) ≥ 1, and set ξ n (t) = δ 0 if N n (t) = 0. Let ν be the probability measure on (0, ∞) with probability density function h(y) = Ai(y + γ 1 ) ∞ 0 Ai(z + γ 1 ) dz .
The appearance of the Airy function here is not a surprise. The Airy function appeared in the early work on traveling waves by Tsimring,Levine,and Kessler [33]. It also arises in the the work of Neher and Hallatschek [27], who studied essentially the same model that we are considering in this paper, as well as in [13], where one of the equations that is central to the work of Neher and Hallatschek [27] had previously been studied. The Airy function also arises in the expression for the position of the right-most particle in branching Brownian motion with inhomogeneous variance, as shown in [26].

A heuristic analysis based on large deviations
ldsec Branching Brownian motion with an inhomogeneous branching rate was also studied in [7,21], where the authors considered the case in which a particle at x branches at a rate proportional to |x| p , where 0 < p < 2. The techniques of proof that we will use in the present paper are quite different from those used in [7,21]. However, while the large deviations techniques used in [7] are not sufficiently precise to prove the main results of this paper, a heuristic calculation based on these techniques provides insight into the behavior of the process and helps to explain the motivation for our proof strategy. We therefore summarize this calculation here, even though it is not logically necessary for understanding the rest of the paper.
Let T be a large time, and let f : [0, T ] → R. According to results in [7], if the process starts with one particle at f (0), then the expected number of particles that stay close to the function f through time T can be approximated by This is because the birth rate minus the death rate for particles that do manage to follow near f will be approximately β n f (u) at time u, and the probability that a Brownian motion with drift −ρ n manages to follow near f is roughly exp( 1 2 T 0 (f ′ (u) + ρ n ) 2 du) by Schilder's theorem. The actual number of such particles will be comparable to the expected number provided that the integral in (1.13), when evaluated from 0 to t, is nonnegative for all t ∈ [0, T ]. Otherwise, at the point that the integral becomes negative, the expected number of particles will be exponentially small and Markov's inequality entails that with high probability no particles will manage to follow the trajectory.
Thinking along these lines, it becomes clear that the position of the right-most particle should roughly follow the trajectory f that makes the integral equal to zero for all t ∈ [0, T ]. In particular, if f (u) = ρ 2 n /2β n for all u ∈ [0, T ], then the integrand in (1.13) is zero. Therefore, if we start with one particle at the position ρ 2 n /2β n at time zero (and the descendants of this particle do not die out quickly), then the right-most particle will stay near the position ρ 2 n /2β n . This calculation explains why we defined L n to be close to ρ 2 n /2β n . We now consider the trajectory f z followed by particles that are near z at time T , starting from one particle at the position ρ 2 n /2β n at time zero. According to Theorem 7 of [7], these particles initially follow the trajectory of the right-most particle, so we have f z (t) = ρ 2 n /2β n for t ∈ [0, t z ] for some t z . Then for u ∈ [t z , T ], equation (18) of [7] implies that f ′′ z (u) = −β n , which means f z (u) = a + bu − (β n /2)u 2 for u ∈ [t z , T ], with the conditions f z (T ) = z, f z (t z ) = ρ 2 n /2β n , and f ′ z (t z ) = 0, for some real numbers a and b. Solving, we get (1.14) tz This means that the number of particles near z at time T is approximately exp(g(z)), where (1.15) gdef We can then calculate which means that for small z, a Taylor expansion gives We then see that at time T , the empirical distribution of particles should be approximately normal with mean zero and variance ρ n /β n , consistent with Theorem 1.1. For this reasoning to be valid, the standard deviation ρ n /β n needs to be much smaller than the distance between zero and the right-most particle, which is ρ 2 n /2β n . This is indeed the case when (1.2) holds. Note also that T − t 0 = ρ n /β n , which explains why, in Theorem 1.1, the Gaussian shape arises after the process has evolved for time ρ n /β n .
Finally, note that although the bulk of the distribution of particles is Gaussian, we see different behavior near the right edge. In particular, writing y = ρ 2 n /2β n − z, from (1.15) we have and since the first term dominates for small y, we see an exponential decay of rate ρ n for the particle density near the right edge. This is partially explained by Theorem 1.2, which describes the configuration of particles that are within a distance of order β −1/3 n from the right edge at L n .

Additional connections to previous work
Branching Brownian motion has previously been used to model populations undergoing selection in [5,9,10,14,25]. However, in these works, it was assumed that the branching rate of a particle does not depend on the position of the particle. Instead, to model selection, particles are killed when they drift too far to the left, that is, when their fitness gets too low. This model leads to substantially different behavior. The empirical distribution of particles is not approximately Gaussian but rather most particles end up close to the left edge. For example, for the model studied in [5] in which particles are killed when they reach the origin, when the system has N particles that are in a nearly stable configuration, the density of particles near y will be roughly proportional to e − √ 2y sin( √ 2πy/ log N ). See [6] for a precise formulation and proof of this result. Nevertheless, the techniques of proof used in this paper are very similar to those used in [5,6].
The main results of [5] state that when the branching rate is constant but particles are killed upon reaching the origin, under suitable initial conditions, the number of particles over time behaves like Neveu's continuous-state branching process, while the genealogy of the particles can be described by the Bolthausen-Sznitman coalescent. The result concerning the genealogy of the particles had previously been predicted in [9,10]. We believe that similar results should hold for the process studied in the present paper. Indeed, (2.14) and Lemma 2.10 below closely resemble Lemmas 11 and 12 of [5] respectively, which are two of the key steps in proving the convergence to Neveu's continuous-state branching process. Also, it was established in [27] by nonrigorous methods that the genealogy of the particles for the process studied in the present paper should be described by the Bolthausen-Sznitman coalescent, following an initial time period in which coalescence does not occur because particles sampled from the bulk of the distribution at time t will most likely be descended from distinct ancestors near the right edge of the distribution at time t − ρ/β. However, in the present paper, we focus on establishing the Gaussian shape for the empirical distribution of particles, and we defer consideration of the genealogy of the particles to a future work.
Beckman [3] also considered branching Brownian motion in which the branching rate depends on the position of the particle. She assumed that at time zero, there are N particles placed independently according to some density, and established a hydrodynamic limit for the evolution of the empirical distribution of particles over time as N → ∞. An important difference between the work in [3] and the present paper is that in [3], time is not rescaled, so the results essentially pertain to what happens at a fixed time t in the limit as N → ∞, whereas here we consider times of the order ρ/β, which tends to infinity. Also, while the work in [3] was likewise motivated by the consideration of evolving populations, the results in [3] have been established only for the case in which the branching rate of the particles is a bounded function of the position.
1.5 Implications for discrete population models iscrete_sec We briefly return here to the discrete population model mentioned at the beginning of the paper, in which there is a fixed population of size N and, at rate µ = µ N , each individual acquires beneficial mutations that increase the individual's fitness by s = s N . We believe that, if the mutation rate is sufficiently large, then the evolution of an individual's fitness over time can reasonably be approximated by Brownian motion, and the results established in this paper should carry over to the discrete model. In particular, the fitness distribution of the population should evolve like a Gaussian traveling wave. Furthermore, because the convergence of random walks to Brownian motion is not sensitive to the step size distribution of the random walk (assuming finiteness of the second moment), we believe that this correspondence should extend to discrete models in which the fitness change resulting from a mutation is random, and mutations could be deleterious as well as beneficial.
Note that in the discrete population model mentioned above, the standard deviation of the number of mutations that an individual gets in one time unit is √ µ N , so the standard deviation of the fitness change of an individual in one time unit is s N √ µ N . In the branching Brownian motion model, the standard deviation of the fitness change of a particle in one time unit is β N , so we have the correspondence Furthermore, (2.6) below states that for the branching Brownian motion model, if (1.6) and (1.7) hold, the number of particles in the system at later times is of the order β Note that (1.19) fails to hold when µ N and s N are both of the order 1/N , in which case the population can be studied using a classical diffusion approximation. The quantity N 3 µ N s 2 N also figures prominently in the work of Good,Walczak,Neher,and Desai [19]. They considered the case in which N µ N → ∞, N s N → 0, and N 3 µ N s 2 N → c ∈ (0, ∞), which led to what they called the "fine-grained coalescent". Their parameter regime would correspond to our model with ρ 3 N /β N → c ∈ (0, ∞), which entails weaker selection than what we consider in this paper. We emphasize, however, that the results in this paper do not rigorously establish any results for the discrete population models. To prove such results, it would be necessary to extend the results in this paper for branching Brownian motion to the case of branching random walks, in which the mutations would correspond to random walk steps. Furthermore, it would be necessary to adapt the analysis to the case of a model of fixed population size, which presents technical challenges because individuals in the population no longer evolve independently. We do not pursue these matters further in this paper.

Outline of the Proofs
The proofs of Theorems 1.1 and 1.2 rely on a combination of first moment estimates based on the many-to-one lemma, second moment estimates to control fluctuations, and careful truncation arguments. We record in this section some of the intermediate results that are important for the argument and defer the more technical proofs until later sections.
We first introduce here some more notation. As mentioned in Section 1, from now on we will drop the subscript n from much of our notation, for example writing ρ and β in place of ρ n and β n , and writing L in place of L n . We emphasize again that is important for the reader to keep in mind that these parameters do depend on n. We have only excluded the subscripts to lighten the burden of notation.
When the initial configuration consists of a single particle at the location x, we denote probabilities and expectations by P x and E x . For real numbers A, we define Note that L A depends on n, although again we omit the subscript. Note also that L 0 = L. We also introduce the shorthand notation For A ∈ R and x ∈ R, we define For A ∈ R and t ≥ 0, we define z n,A (X i,n (t)).
Note that Z n (t) = Z n,0 (t). More generally, given A ∈ R and a function ϕ : We introduce L A above because it will sometimes be necessary to consider a modification of the process in which particles are killed when they reach L A . In this case, it will be important to keep track of how various constants depend on A. Given two sequences (a n ) ∞ n=1 and (b n ) ∞ n=1 , when we write a n b n or b n a n , the ratio a n /b n must be bounded above by a positive constant that does not depend on A. We write a n ≍ b n to mean that both a n b n and a n b n hold. Throughout the rest of the paper, C k for a nonnegative integer k will denote a fixed positive constant. The value of C k may not depend on n or A and does not change from one occurrence to the next.

The empirical distribution of particles
A large part of our proofs will consist of detailed first and second moment estimates, allowing us to approximate the density of particles in different regions of space. We will detail these in sections 2.2 to 2.7 below. Putting these moment estimates aside, there are three main steps required to prove Theorems 1.1 and 1.2. The first step requires considering the configuration of particles near the right edge. As long as the initial configuration of particles satisfies (1.6) and (1.7), a short time later the particles near the right edge should settle into a relatively stable configuration, described by the density h defined in (1.12). For particles within a distance of order β −1/3 from the right edge, it should take a time of order β −2/3 for the particles to reach this relatively stable configuration. For technical reasons, to make it easier to control particles that drift a bit further than order β −1/3 away from the right edge, we give the particles a bit more time to move into this configuration and establish the result below for times much larger than β −2/3 log(ρ/β 1/3 ). The proof of Proposition 2.1, whose statement is identical to Theorem 1.2 except that we insist that t n ≪ ρ/β, uses techniques similar to those used in [6], and is given in section 5.
The second step involves considering the configuration of particles near the origin. As long as the initial configuration of particles satisfies (1.6) and (1.7), the particles near the origin at time approximately ρ/β, most of which will have been descended from particles that were near L at time zero, should be approximately in a Gaussian configuration. Note that (2.6) below, while not strictly required for the proof of Theorem 1.1, provides useful insight into the behavior of the process, as it says that the number of particles in the system at time approximately ρ/β is determined, to a high degree of precision, by the value of Z n (0). The proof of Proposition 2.2, which also uses techniques similar to those used in [6], is given in section 6.
The third step is to show that if (1.6) and (1.7) hold, then these conditions will still hold with high probability at later times, which are order ρ/β in the future. This is established in the following proposition, which is proved in section 7. YZmain Proposition 2.3. Suppose the initial configuration of particles satisfies (1.6) and (1.7). Suppose the times t n are chosen so that lim n→∞ βt n ρ = τ ∈ (0, ∞).
Then, with probability tending to one as n → ∞, the conditions (1.6) and (1.7) hold with Z n (t n ) and Y n (t n ) in place of Z n (0) and Y n (0) respectively.
We now show how Propositions 2.1, 2.2, and 2.3 imply Theorems 1.1 and 1.2. Because these proofs involve subsequence arguments, we will return to writing ρ n and β n instead of ρ and β to avoid confusion.
Proof of Theorem 1.1. It suffices to show that every subsequence (n j ) ∞ j=1 contains a further subsequence (n j k ) ∞ k=1 for which ζ n j k (t n j k ) ⇒ µ as k → ∞. By (1.8), the sequence β n t n ρ n − 1 ∞ n=1 is bounded and non-negative for large n. Therefore, given a subsequence (n j ) ∞ j=1 , we can choose a further subsequence (n j k ) ∞ k=1 for which If τ = 0, then (2.5) holds along this subsequence, and it follows immediately from Proposition 2.2 that ζ n j k (t n j k ) ⇒ µ as k → ∞. Suppose instead that τ > 0. Choose any sequence (s n ) ∞ n=1 for which (2.5) holds with s n in place of t n , and let u n = t n − s n . Note that lim k→∞ β n j k u n j k ρ n j k = τ ∈ (0, ∞), so Proposition 2.3 implies that (1.6) and (1.7) hold with Z n j k (u n j k ) and Y n j k (u n j k ) in place of Z n (0) and Y n (0). We can therefore apply the Markov property at time u n j k , followed by Proposition 2.2 with s n in place of t n , to see that ζ n j k (t n j k ) ⇒ µ.
Proof of Theorem 1.2. The proof is very similar to the proof of Theorem 1.1. It suffices to show that every subsequence (n j ) ∞ j=1 contains a further subsequence (n j k ) ∞ k=1 for which ξ n j k (t n j k ) ⇒ ν as k → ∞. By (1.10), the sequence (β n t n /ρ n ) ∞ n=1 is bounded. Therefore, given a subsequence (n j ) ∞ j=1 , we can choose a further subsequence (n j k ) ∞ k=1 for which lim k→∞ β n j k t n j k If τ = 0, then (2.4) holds along this subsequence, and it follows from Proposition 2.1 that ξ n j k (t n j k ) ⇒ ν. Suppose instead that τ > 0. Choose any sequence (s n ) ∞ n=1 for which (2.4) holds with s n in place of t n , and let u n = t n − s n . Note that lim k→∞ β n j k u n j k ρ n j k = τ ∈ (0, ∞), so Proposition 2.3 implies that (1.6) and (1.7) hold with Z n j k (u n j k ) and Y n j k (u n j k ) in place of Z n (0) and Y n (0). We can therefore apply the Markov property at time u n j k , followed by Proposition 2.1 with s n in place of t n , to see that ξ n j k (t n j k ) ⇒ ν.

2.2
The density for the unkilled process nkilled_sec As mentioned above, a large amount of the work in our proofs involves moment estimates that allow us to bound the number of particles in certain regions of space. We begin with first moment calculations. We denote by p t (x, y) the density for the process, which means that if there is one particle at x at time zero, then the expected number of particles in the Borel set D at time t is given by To calculate the density, we can invoke the many-to-one lemma, which is proved, for example, in [20]. To do this, we first compute the density for the process when ρ = 0, which we denote by q t (x, y). Let (B t ) t≥0 be one-dimensional Brownian motion started at B 0 = x. The many-to-one lemma states that if f : R → R is a nonnegative measurable function, then (2.7) many1 Consequently, the density for the process without drift can be read from formula 1.8.7 on page 141 of [11], which yields The drift of −ρ can be added using a standard Girsanov transformation, which implies that p t (x, y) = e ρ(x−y) e −ρ 2 t/2 q t (x, y) and therefore (2.8) ptxy Integrating (2.8) with respect to y gives Alternatively, one can obtain (2.9) by applying the many-to-one formula (2.7) to Brownian motion with drift, which gives The result (2.9) then follows from equation 1.8.3 on page 141 of [11], which states that

Brownian motion killed at a random time alminen_sec
In order to control accurately the number of particles in certain regions of space, we will need to use moment estimates for a process where some of the particles are killed upon hitting a barrier.
To carry out these calculations we will need estimates on Brownian motion killed at a random time, which we collect here.
Suppose that x > 0 and (B t ) t≥0 is a Brownian motion started from x. We will use an interpretation of the integral t 0 β|B u |du as a random clock, following Salminen [31]. Let the first time that the Brownian motion hits zero. Let E be an independent exponential random variable of parameter 1, and define where Υ is some graveyard state. Then for any Borel set A ⊆ (0, ∞), (2.10) killedreprese Let τ 0 = inf{t ≥ 0 :X t = 0}. Writingp t (x, y) for the transition density ofX t and π x (t) for the density of τ 0 , we will need the following two results from [31] (noting that the value of β used in [31] differs from ours by a factor of 2 and the densities in [31] are written with respect to twice Lebesgue measure): which is [31, Proposition 3.9], and which is [31, (3.2)].

The density for the killed process
Returning to our branching Brownian motion, we will often need to consider a truncated version of the process, in which particles are killed as soon as they surpass some level ℓ ∈ R. For x < ℓ and y < ℓ, we denote the density for this process by p ℓ t (x, y). Defining to be the first hitting time of ℓ by our Brownian motion B t , by the many-to-one formula we have Making the transformation B ′ u = ℓ − B u , and setting T ′ 0 = inf{t ≥ 0 : B ′ t ≤ 0}, we see that . Recognising the last expectation as the transition density of the killed Brownian motion from Section 2.3, from (2.11) we have (2.13) ptKeq We will typically take ℓ = L A for some real number A. The exponent (β(ℓ + (2β) −1/3 γ 1 ) − ρ 2 /2)t in the leading term in (2.13) is zero when ℓ = L, which is why L is the correct level at which to kill particles to keep the number of particles in the system approximately stable over time.
As a consequence of the formula (2.13) for the density, it is possible to show that for any A ∈ R, if we consider a modified process in which particles are killed when they reach L A , then for all t ≥ 0 and x < L A , we have (2.14) zmeaneq In particular, the process (Z n (t), t ≥ 0) is a martingale. This can be proved using (2.13), the dominated convergence theorem, and the following orthogonality relation from Section 4.4 of [34]: We will not use the fact that (Z n (t), t ≥ 0) is a martingale in the rest of the paper, so we do not include the full proof here, but it may provide insight into why (Z n (t), t ≥ 0) is an important measure of the "size" of the process.

Approximate density formulas approxden
We will need to establish some approximations to the density formulas (2.8) and (2.13). We will therefore state in this subsection six lemmas, all of which will be proved in section 3. As indicated in section 1.3, most particles that are alive at time t will be descended from particles that were near L at time t − ρ/β. Therefore, it will be useful to have the following approximation for p t (x, y), which holds when x ≈ ρ 2 /2β and t ≈ ρ/β. Note that here and throughout this subsection and the next two, the time t implicitly depends on n.
If |w| ≪ ρ/β and 0 ≤ s ≪ ρ 1/4 β −3/4 , then It follows from (1.2) that if |y| ρ/β, then the last four terms inside the exponential in (2.16) are all o(1). Note also that Lemma 2.4 indicates that for x sufficiently close to ρ 2 /2β, the expected number of descendant particles in a particular set at time t closely matches the Gaussian distribution with mean 0 and variance ρ/β.
Our remaining approximations pertain to the process in which particles are killed when they reach L A . As long as t is large enough, and x and y are sufficiently close to L A , the right-hand side of (2.13) can be approximated by its leading term.
In particular, if there exists a strictly positive constant C such that We will also need several additional formulas that can be used when (2.18) is not satisfied. Lemma 2.6 is most useful when t ≤ ρ −2 , while Lemma 2.7 is useful for slightly larger values of t, particularly when both x and y are close to L A . Lemma 2.8 is useful when x is close to L A , and y is far enough away from L A that a particle going from x to y is unlikely to be affected by the right boundary at L A , unless it hits the boundary almost immediately. dapprox1 Lemma 2.6. For all t ≥ 0, ℓ ≥ 0, x < ℓ, and y < ℓ, we have The next result gives an estimate for the integral of the density for branching Brownian motion when the particles are killed at L A and the initial particle at x is close to the right boundary. The result can be compared to the formula (2.9) for the process without killing.
where 0 ≤ s ≪ t, and suppose A ∈ R and x < L A . Then

Second moment estimates
To control the fluctuations in the process, we will need good second moment bounds. Given A ∈ R and t ≥ 0, let There is nothing particularly special about these functions. We will need to split the integrals that arise in our second moment bounds into several cases, and these functions will give convenient points at which to split, taking into account condition (2.18) in Lemma 2.5. Let ϕ : (−∞, L A ) → R be a measurable function. Lemma 2.10 establishes a second moment bound for the quantity V ϕ,n,A (t) defined in (2.3). This bound will help us to control the fluctuations of the number of particles at time t in the interval (K A (t), L A ). The particles outside this interval will be controlled by other methods. Note that in Lemma 2.10, the time t and the function ϕ are implicitly allowed to depend on n.
2momprop Lemma 2.10. Fix A ≥ 0, and let ϕ : (−∞, L A ) → R be a bounded measurable function such that ϕ(y) = 0 unless K A (t) < y < L A . Consider the process in which there is initially one particle at x and particles are killed when they reach L A . Suppose K A (t) < x < L A , and suppose t β −2/3 . Then To prove Theorem 1.1 and establish that the empirical distribution of particles is asymptotically Gaussian, we will also need to control the fluctuations in the number of particles close to the origin after a time that is approximately ρ/β. mainvarg Lemma 2.11. Consider the process in which there is initially one particle at x and particles are killed when they reach L.
Because the proofs of Lemmas 2.10 and 2.11 are rather tedious, we defer them until section 8.

Estimates for the particles that reach L
We estimate here the rate at which particles reach the right boundary at L A , when particles are killed upon hitting this level. We will begin by considering a more general right boundary at ℓ, before we later specialise to ℓ = L A . Suppose we start with one particle at x < ℓ and kill particles when they hit level ℓ. For 0 ≤ u < v, let r ℓ x (u, v) be the expected number of particles that hit ℓ between times u and v. Letr ℓ x (t) denote the rate at which particles hit ℓ at time t, so that From the many-to-one lemma, recalling that we defined Recognising the last expectation as the density of the hitting time of zero of the killed Brownian motion introduced in Section 2.3, and recalling the notation of that section, from (2.12) we havẽ However, from (2.10), Therefore, using the many-to-one lemma again, Since we know that p ℓ t (x, ℓ) = 0, and Lemmas 2.5 and 2.7 give us upper bounds on p ℓ t (x, ℓ − h) for h > 0, from (2.24) we obtain the following corollary. rtedge Corollary 2.12. For all t ≥ 0, ℓ > 0 and x < ℓ, we havẽ If ℓ = L A for some fixed A ∈ R and, in addition, there exists a positive constant C > 0 such that We will also use the following lemma to estimate the number of descendants of a particle at x that reach L A . This estimate involves bounding separately the expected number of descendants that hit L A during an initial time period, for which we can use the bound in (2.25), and the number of descendants that hit L A later, for which the bound in (2.27) is valid. This result will be proved in section 4.
Finally, we return to the original process in which particles are not killed when they reach L A . Lemma 2.14 below shows that the probability that a descendant of a particle that reaches L A will survive for a reasonably long time is of the order ρ 2 . Note that because a particle near L A has an effective branching rate of b(x) − d(x) ≈ ρ 2 /2, this result is to be expected in view of classical results on the survival probability of Galton-Watson processes. A complication is that the birth rate changes as the particles move. Note that in [5,8], stronger results were obtained related to the number of surviving descendants of particles that reached the right boundary, and these results were essential for establishing convergence to a continuous-state branching process. However, the weaker result established in Lemma 2.14 will be sufficient for our purposes. Lemma 2.14 will also be proved in section 4.
Lsurvive Lemma 2.14. Suppose at time zero, there is a single particle at There is a positive constant C 1 such that for sufficiently large n, the probability that any individual

Proofs of density approximations densec
In this section, we prove the results stated in section 2.5, which establish approximate formulas for the densities p t (x, y) and p L A t (x, y).

Facts about Airy functions
In this subsection we collect some facts about Airy functions which will be needed later in the paper. In particular, Lemma 3.1 below will be important for the proof of Lemma 2.5. We first record some facts which can be found in [34]. Below C 2 , C 3 , C 4 , and C 5 are positive constants. Using ∼ to indicate that the ratio of the two sides tends to one, we have . We will also use that which can be deduced from equation (2.46) in [34] and the continuity of the derivative of the Airy function.
Airyratlem Lemma 3.1. For all z ≥ 0 and k ∈ N, we have Proof. Let A be a positive constant. If 0 ≤ z ≤ A, then it follows from (3.2) and (3.5) that which is stronger than (3.7).
3.2 Proofs of Lemmas 2.5, 2.6, 2.7, and 2.8 We begin by using Lemma 3.1 to provide the necessary error estimates to prove Lemma 2.5.
Proof of Lemma 2.5. The expression (2.17) with E A = 0 is the k = 1 term from (2.13), as can be seen by recalling (2.1). Denote by r k (t, x, y) the ratio of the kth term in (2.13) to the first term, when Using (3.3) to bound the first factor and Lemma 3.1 to bound the second factor, we get which implies (2.17). We then obtain (2.19) and the last conclusion of the lemma by estimating E A (t, x, y) using (3.2).
The proof of Lemma 2.6 uses the fact that no particles go above ℓ to bound the branching rate, but otherwise uses the unkilled process. This explains the fact that it is similar to, but not the same as, (2.8).
Proof of Lemma 2.6. Fix a < b ≤ ℓ. Let N t (a, b) denote the number of particles in (a, b) at time t that have never hit ℓ. Let (B s ) s≥0 be Brownian motion started at x. By applying the many-to-one formula (2.7) to Brownian motion with drift, we get as claimed.
The proof of Lemma 2.7 uses a trivial bound on the branching rate, but uses the killed process as opposed to the unkilled process.
Proof of Lemma 2.7. We proceed as in the proof of Lemma 2.6 but keep the restriction that B s − ρs < ℓ for all s ≤ t, and apply Girsanov's theorem followed by the reflection principle. This Applying the elementary bound which yields the result.
Finally, the proof of Lemma 2.8 is more involved and combines the formula (2.8) with the estimate in Lemma 2.7.
Proof of Lemma 2.8. Let s = β −2/3 . We apply the Chapman-Kolmogorov equation at time s and then use Lemma 2.7 to bound p L A s (x, z) and (2.8) to bound p t−s (z, y), which gives After a few lines of algebra, we obtain Therefore, substitutingz in place of z in the integral in (3.14), we get Because we are assuming Using (3.16), (3.17), (3.18), and (3.19) to simplify the exponential terms in (3.15), we get For z ∈ R, we will write z + = max{0, z} for the positive part of z. It is easy to check that if Z has a normal distribution with mean µ ∈ R and standard deviation σ > 0, then Because Because s −3/2 = β and t − s ≍ t, the lemma follows from (3.20) and (3.21).

Proof of Lemma 2.4
From (2.8), we have Also, Next, we observe that To estimate the first term in (3.26), we note that the terms in the infinite sum with k ≥ 4 are small due to (1.2) and the assumption that s ≪ ρ 1/4 β −3/4 . Therefore, To estimate the second term in (3.26), we again use (1.2) and the hypotheses on |w| and s to get For the third term in (3.26), we have We can now evaluate the right-hand side of (3.22) by putting together the results in (3.23), (3.24), (3.25), (3.27), (3.28), and (3.29), which yields (2.16).

Proof of Lemma 2.9
Because p L A t (x, y) ≤ p t (x, y), it follows from (2.9) that it suffices to prove the result when L A −x ≤ β −1/3 , which we will assume for the rest of the proof. Note that Combining this estimate with Lemma 2.8, and separating out the terms involving y in the exponential factor, we get Note that if a ∈ R, then (y − x) 2 + ay = (y − (x − a/2)) 2 + ax − a 2 /4. Applying this result to the second exponential factor above with (3.30) expnoy After some tedious but straightforward algebra, we see that the exponential factor on the second line of (3.30) can be written as Now using that t = ρ/β − s, and that x = ρ 2 /2β + O(β −1/3 ) by the definition of L A and the assumption that L A − x ≤ β −1/3 , we can write the second exponential factor in (3.31) as Substituting the results in (3.31) and (3.32) back into (3.30), using that 1/(β 2/3 t) → 0, and integrating with respect to y, we get The integral in the previous line is bounded above by where V has a normal distribution with mean x − a/2 and variance t. This is the same as where W has a normal distribution with variance and mean We therefore have E[max{1, W }] max{1, β 2/3 s} exp(β 2/3 s/2). Because this expression gives an upper bound for the integral in (3.33), the result follows.

Particles that hit L A rtsec
In this section, we will prove Lemmas 2.13 and 2.14. Lemma 2.13 pertains to the process in which particles are killed when they reach L A , and provides an estimate for how many particles are killed. Lemma 2.14 pertains to the process in which particles are allowed to continue after reaching L A . It provides an estimate for the probability that the descendants of a particle will survive for a significant period of time after the particle reaches L A . We also prove Lemma 4.3 below, which bounds the expected contribution to Y n (t), for small times t, from an initial particle at L A . We begin with the following simple integral estimate. 32int Lemma 4.1. If a > 0 and b > 0, then Proof. Make the substitution y = b 2 /ax and then recall that Γ(1/2) = √ π.
Proof of Lemma 2.13. Define Let 0 < δ < 2 −1/3 . It follows from Corollary 2.12 and the fact that γ 1 < 0 that It follows from Lemma 4.1 that Therefore to prove (4.2) it suffices to show that We consider two cases. First, suppose t x,A ≤ δ −1 β −2/3 . Then (4.4) holds because the left-hand side is bounded above by a positive constant, while the right-hand side is bounded below by a positive constant because t x,A ≥ 2 1/3 β −2/3 by (4.1). Next, suppose t x,A ≥ δ −1 β −2/3 . It follows from (4.1) that and therefore By choosing δ sufficiently small, we see that this bound, combined with the fact that the function y → −ay 3 + by is bounded above for all a > 0 and b > 0, implies (4.4) and therefore (4.2) holds.
In particular this proves the lemma in the case s ≤ t ≤ t x,A .
Next, suppose that t x,A ≤ s ≤ t. Note that t x,A has been defined so that (2.26) holds with equality when t = t x,A and C = 1. If s ≤ u ≤ t, then (2.26) holds with u in place of t. Therefore, by Corollary 2.12, and combine the bounds from (4.2) and (4.5).
Before we prove Lemma 2.14, we show that our process cannot explode in finite time.
noexplosion Lemma 4.2. Suppose that, at time zero, there is a single particle at x. Then for all t > 0, the random variable Proof. Fix k > x and consider a system where particles are frozen (that is, they no longer move or branch) once they hit level k. Call the resulting probability measureP. Let A be the number of particles that have been frozen by time t. By the many-to-one lemma, where underP, the process (B u ) u≥0 is a Brownian motion with drift −ρ that is frozen upon hitting the level k. Since B u ≤ k for all u underP, we deduce that Now, the probability that B t = k is exactly the probability that a Brownian motion with drift −ρ hits k before time t, which is smaller than the probability that a Brownian motion with no drift hits k before time t. It is well-known that the first hitting time of level y by a standard Brownian motion (W t ) t≥0 started from 0 is equal in distribution to (y/W 1 ) 2 ; thus, using a standard Gaussian approximation, which converges to 0 as k → ∞. The result follows.
Proof of Lemma 2.14. Fix δ > 0. Let K m = x(1 + δ) m for each m ≥ 0. Label the initial particle at x to be type 0. Whenever a particle reaches K m for the first time, it becomes type m. The type of a particle is never allowed to decrease, so a particle will have type m if at some time it had an ancestor above K m , but it never had an ancestor above K m+1 . When a birth occurs, offspring have the same type as the parent. For nonnegative integers m, let Let D 0 be the event that there is a type 0 individual alive in the population at time T 0 . For positive integers m, let D m be the event that there are type m individuals in the population continuously from time T m−1 until time T m . Let D be the event that some individual survives until time T . We claim that To see this, note that if D 0 fails to occur, then there are no type 0 individuals left at time T 0 , but there could be individuals that migrated to the right of K 1 and became type 1 individuals. If D 1 also fails to occur, then the type 1 individuals must all be gone by time T 1 , but there could be individuals that became type 2. Repeating this argument, we see that if none of the D i occur, then there cannot be individuals of any type remaining at time T . The only further possibility is that there are individuals alive at time t that have had type j + 1 by time T j , for all j ∈ N. By Lemma 4.2, this has probability zero.
We therefore aim to bound the probability of D m . Suppose m ≥ 1. For any u ≥ 0, by (2.25), Therefore, using also that ρ 2 u/2 ≥ 0, we havẽ It follows from this bound and Lemma 4.1 that the expected number of particles to hit K m by time T m−1 , if particles are killed upon reaching K m , is at most Now suppose, for m ≥ 1, a particle reaches K m before time T m−1 . For m = 0, we can consider instead the particle at x at time zero. Particles of type m have positions x ≤ K m+1 and therefore effective branching rate b(x) − d(x) ≤ βK m+1 . For a continuous-time branching process in which each individual gives birth at rate λ and dies at rate µ, it is well-known that the probability that the population survives for at least time t is given by .
This formula can be deduced, for example, from results in Section 5 in Chapter III of [1]. For any µ > 0 and t > 0, one may check that the derivative of the function z → z+µ(1−e −zt ) z is always negative and therefore the function is decreasing in z. Thus the function is increasing in z. Applying this with z = λ − µ ≤ βK m+1 , µ ≥ ∆ (from (1.4)) and t = τ m , we see that the probability that a particle that reaches K m before time T m−1 has descendants of type m alive in the population at time T m is bounded above by .
Proof. By (2.8), Because t ρ −2 , the terms ρ 2 t/2, βL A t/2, and β 2 t 3 /24 are all bounded above by positive constants. It follows that  In this section, we will prove Proposition 2.1, which gives a precise description of the density of particles near the right edge. Because it will sometimes be necessary to condition on the initial configuration of particles, we will define (F t , t ≥ 0) to be the natural filtration associated with the branching Brownian motion process. We begin with the following lemma, which states that when (1.7) holds, particles that start out close to L can be neglected because they will not have descendants surviving for very long.
Then the probability that some particle that is to the right of L A at time zero has a descendant alive in the population at time 2C 1 ρ −2 tends to zero as n → ∞.
Proof. First note that any particle that is to the right of ρ/β at time zero contributes at least e ρ 2 /β ≫ ρ −2 e ρL to Y n (0). It therefore follows from (1.7) that the probability that any particle is to the right of ρ/β at time zero tends to zero as n → ∞, and we can restrict our attention to particles that start to the left of ρ/β. Fix ε > 0. Say that a particle at time 0 "survives" if it has a descendant alive at time 2C 1 ρ −2 . Let S be the number of particles whose positions at time 0 are between L A and ρ/β and who survive; and define E 0 = E[S|F 0 ]. Then, using the conditional Markov inequality, Since ε > 0 was arbitrary, it therefore suffices to show that P(E 0 > ε) → 0 as n → ∞.
By Lemma 2.14, a particle i with X i (0) ∈ [L A , ρ/β] survives with probability at most 2βX i (0)/∆. (In fact Lemma 2.14 bounds the probability that a particle has a descendant alive at time C 1 /βX i (0), but this is smaller than 2C 1 ρ −2 when n is large since X i (0) ≥ L A and L A ≥ ρ 2 /2β for large n.) Thus We now use the elementary bound x ≤ e x together with (1.2) to say that, for large n, The first sum on the right-hand side tends to 0 in probability directly by (1.3) and (1.7). The second also tends to 0 in probability, since which converges to 0 as n → ∞ by (1.3) and (1.7). This completes the proof.
We now introduce some additional notation. Recall from (2.21) that K A (t) = L A − βt 2 /66. We also define We set H(t) = H 0 (t) and K(t) = K 0 (t). Note that if H A (t n ) ≤ x < L A and H A (t n ) ≤ y < L A , then as long as t n ≫ β −2/3 , the condition (2.20) holds, and therefore Lemma 2.5 can be used to estimate p L A tn (x, y). When K A (t n ) < x < L A and K A (t n ) < y < L A , Lemma 2.10 can be used for second moment bounds. Other methods are needed to control the contribution from particles to the left of H A (t). The next lemma will be very useful in this regard.
If x < L A , 0 ≤ ζ ≤ βt n /2, and β −2/3 ≪ t n ≪ ρ/β, then Proof. Suppose x ≤ L A − 1 9 βt 2 n . By (2.8) and the fact that p L A tn (x, y) ≤ p tn (x, y), we have Because β 2 t 3 n ≫ β 2/3 t n and β 2 t 3 n ≫ βt n /ρ, it follows that (5.3) holds. To establish (5.4), we use the same argument. Instead of having x ≤ H A (t n ) and y ≤ L A , we now have x ≤ L A and y ≤ H A (t n ). However, the resulting bound on β(y + x)t n /2 is the same, and the rest of the calculation proceeds identically.
Proof of Proposition 2.1. Let g : R → [0, ∞) be a bounded measurable function, and let On the event that N n (t) ≥ 1, let Otherwise, let Φ n (g) = 0. Let Φ n (1) be the value of Φ n (g) when g(x) = 1 for all x. Then, when N n (t) ≥ 1, we have We will show that for all κ > 0, we have It will then follow that for all κ > 0, we have which by, for example, Theorem 16.16 of [23] will imply the statement of the proposition. It therefore remains to prove (5.8). We will estimate Φ n (g) by dividing it into seven pieces, depending mostly on the location of the particle at time t n and the location of the ancestral particle at time zero. For i ∈ {1, . . . , N n (t)} and s ∈ [0, t], let a i,n (s, t) be the location at time s of the ancestor of the ith particle at time t. We partition the particles at time t n into the following seven subsets: S 3,n = i / ∈ S 1,n ∪ S 2,n : a i,n (0, t n ) < H(t n ) , S 4,n = i / ∈ S 1,n ∪ S 2,n ∪ S 3,n : X i,n (t n ) < H(t n ) , and note that Φ n (g) = Φ 1,n (g) + · · · + Φ 7,n (g). We will show that the first six terms contribute little to the sum, while the seventh is highly concentrated around its mean. The first term Φ 1,n (g) accounts for the contributions of particles that reach L before time t n − 2C 1 ρ −2 . By Lemma 5.1, with probability tending to one as n → ∞, no particles above L at time zero will have descendants alive at time t n . Consider the process in which particles are killed upon reaching L, and let R n (s, t) be the number of particles killed between time s and time t. By Lemma 2.13, Therefore, using the assumptions (1.6) and (1.7) and the fact that t n ≪ ρ/β, we obtain that as n → ∞, ρ 2 E[R n (0, t n − 2C 1 ρ −2 )|F 0 ] ρ 2 e −ρL Y n (0) + ρ 2 e −ρL β 2/3 t n Z n (0) → p 0.
In view of Lemma 5.1, it follows that with probability tending to one as n → ∞, no particle that hits L before time t n − 2C 1 ρ −2 has descendants alive in the population at time t n . That is, we have lim n→∞ P(Φ 1,n (g) = 0) = 1.

(5.18) Pphi6
Finally, we consider the term Φ 7,n (g). Using Lemma 2.5 in the first step, making the substitution z = (2β) 1/3 (L − y) in the second step, and using that β 4/3 t 2 n → ∞ in the third step, we obtain Therefore, in view of (1.6), we have that for all η > 0, Moreover, using the independence of the descendants of different particles along with Lemma 2.10, we get Therefore, by the conditional Chebyshev's Inequality, for all η > 0 we have The first term on the right-hand side converges in probability to zero by (1.7), and because t n ≪ ρ/β, the second term on the right-hand side converges in probability to zero by (1.6). It follows that Combining (5.15), (5.19) and (5.20) gives that for all η > 0, Finally, combining (5.9), (5.12), (5.14), (5.16), (5.18), and (5.21), we get that for all η > 0, The result (5.8), and therefore the statement of the proposition, now follows from (1.6).

Proof of Proposition 2.2 gausssec
In this section, we prove Proposition 2.2, which shows that when (1.6) and (1.7) hold, the empirical distribution of particles at time approximately ρ/β is asymptotically Gaussian. We begin by proving the following simple lemma concerning the Airy function.
Because I(1) = 1, it will follow from (6.12) that as n → ∞, we have which by, for example, Theorem 16.16 of [23] is enough to imply that ζ n (t n ) ⇒ µ. It remains, then, to prove (6.12). Let η > 0, and recall the definitions of the positive constants C 7 and C 8 from before the statement of Lemma 6.2. Let C 9 be a positive constant chosen large enough that if Z has a standard normal distribution, then P (|Z| > C 9 ) < η.
Because the times t n satisfy (2.5), we have 14) unbound and therefore the configuration of particles at time u n satisfies the conclusions of Proposition 2.1. To prove (6.12), we will follow the trajectories of the particles between times u n and t n . Recalling (5.2), we first partition the particles at time u n into the following four subsets: We then partition the particles at time t n into six subsets. For j ∈ {1, 2, 3}, we define S j,n = {i : a i,n (u n , t n ) ∈ G j,n }.
We also define S 4,n = i : a i,n (u n , t n ) ∈ G 4,n and X i,n (t n ) / ∈ − C 9 ρ/β, C 9 ρ/β , S 5,n = i : a i,n (u n , t n ) ∈ G 4,n , i / ∈ S 4,n , and a i,n (s, t n ) ≥ L for some s ∈ (u n , t n ) , S 6,n = i : a i,n (u n , t n ) ∈ G 4,n and i / ∈ S 4,n ∪ S 5,n .
Then Ψ n (g) = Ψ 1,n (g) + · · · + Ψ 6,n (g). We will show that with high probability, the values of Ψ j,n (g) are small for j ∈ {1, . . . , 5}. The dominant contribution comes from Ψ 6,n (g), which is highly concentrated around its expectation. The term Ψ 1,n (g) accounts for the particles that are above L at time u n . To bound this term we can use the argument leading to (5.9), with u n in place of t n − 2C 1 ρ −2 , to see that with probability tending to one as n → ∞, no particle that either starts above L or reaches L before time u n has descendants alive past time u n + 2C 1 ρ −2 . Because u n + 2C 1 ρ −2 ≤ t n for sufficiently large n, it follows that lim n→∞ P(Ψ 1,n (g) = 0) = 1. (6.15) Ppsi1 We next consider Ψ 2,n (g), which accounts for particles that are below H(u n ) at time u n . If there is a particle at x at time u n , then by (2.9), the expected number of descendants of this particle alive at time t n is Using that t n − u n = (ρ/β) − s n , after a few lines of algebra we get that the expression in (6.16) is equal to It follows that Note that (6.14) implies that s n ≪ u n and therefore βs n ≤ βu n /2 for sufficiently large n, so it follows from (5.6) that and therefore Because u n ≫ ρ 2/3 /β 8/9 by (6.14), we have ρ 2 s n ≪ β 2 u 3 n . Therefore, Combining (6.19) with (1.6) and (1.7) along with the conditional Markov's inequality, we obtain The reasoning leading to (6.18) gives We can write G 3,n = G * 3,n + G * * 3,n , where for i ∈ G 3,n , we say i ∈ G * 3,n if a i,n (0, u n ) ≤ H(u n ) and i ∈ G * * 3,n otherwise. By (5.5), E i∈G * 3,n e (ρ−βsn)X i,n (un) F 0 ≪ e −β 2 u 3 n /73 Y n (0). (6.22) G* When H(u n ) < x < L and H(u n ) < y < L, we can estimate p L un (x, y) using Lemma 2.5, and the error term E 0 (u n , x, y) will be o(1). Therefore, for any constant C 10 > 2 1/3 /(Ai ′ (γ 1 )) 2 , we have for sufficiently large n, e −βsny β 1/3 α(L − y) dy.

) Psi56upper
To obtain the corresponding lower bound, note that by (6.13), Also using again that θ n → 0 uniformly as n → ∞, we get The term Ψ 5,n (g) accounts for the particles that reach L between times u n and t n . We now bound the contribution from this term individually. Take v ∈ [0, t n − u n ], and recall the definition ofr L x (v) from the beginning of section 2.7. From Corollary 2.12 and the fact that (ρ 2 /2) − βL = 2 −1/3 β 2/3 γ 1 , there is a positive constant C 11 such that Now let m n (v) denote the expected number of descendants in the population at time t n of a particle that reaches L at time u n + v. It follows from (2.9) that Because t n − u n = (ρ/β) − s n , a short computation gives It follows that Therefore, using that γ 1 < 0 and (v + s n ) 3 ≥ s 3 n , we obtain from (6.29) and (6.30) that We now integrate over v and apply Lemma 4.1 to see that if there is one particle at x at time u n , then the expected number of particles alive at time t n whose trajectory crosses L between times u n and t n is bounded above by It follows that E[Ψ 5,n (g)|F un ] ≤ C 11 √ 2π g e −C 3 7 /6 e −ρ 3 /3β Y n (u n ).

YZsec
To prove Proposition 2.3, we essentially show that Y n (t n ) and Z n (t n ) remain of the order β 1/3 ρ −3 e ρL after the process has evolved for a time that is of the order ρ/β. For the upper bounds, only truncated first moment estimates, in combination with Markov's inequality, are needed. However, killing particles when they hit L is not sufficient because doing so would kill some particles whose descendants would otherwise contribute significantly to the process at time t n . Therefore, we instead have to kill particles when they hit L A for A < 0, and then move the barrier further away as time increases-that is, make A more negative as a function of time-thereby reducing the number of particles that hit the wall.
Obtaining a lower bound on Z n (t n ) requires second moment estimates. To obtain adequate second moment estimates, the value of A needs to be chosen so that the wall at L A moves closer to the origin at regular intervals. An alternative to this approach would be to follow the techniques in [5,8], which would likely yield the stronger result that (Z n (t), t ≥ 0) converges to a continuous-state branching process. However, the simpler arguments given here are sufficient for our purposes.

Upper bounds on contributions of particles remaining below L A
Recall that a i,n (s, t) is the position of the ancestor at time s of the ith particle at time t, and so that Y * n,A (t) and Z * n,A (t) only count particles that have remained below the level L A for all times s ≤ t. Before we introduce our moving barrier, a large part of our upper bound follows relatively easily from estimates on Y * n,A (t n ) and Z * n,A (t n ) for fixed A < 0. The following fact about the Airy function will help us to compare the values of z n,A (x) for different values of A. AiryZZA Lemma 7.1. If x > 0 and 1/2 < r < 1, then Ai(x + γ 1 ) ≤ 2Ai(rx + γ 1 ).
Proof. We consider three cases. First, suppose 0 < x ≤ −γ 1 . Because Ai(x) > 0 for all x > γ 1 and the Airy function solves the differential equation Ai ′′ (z) = zAi(z), the second derivative Ai ′′ (z) is negative for all z ∈ (γ 1 , 0). Therefore, since r < 1, the average value of the derivative of the Airy function between γ 1 and γ 1 + x is less than the average value of the derivative of the Airy function between γ 1 and γ 1 + rx. The conclusion of the lemma follows from this observation because r ≥ 1/2. Next, let a ′ 1 < 0 be the largest zero of the derivative of the Airy function, which is also the point at which the Airy function attains its maximum. Suppose x > −γ 1 but rx < a ′ 1 − γ 1 . Then, we can apply the result from the previous case with a ′ 1 − γ 1 in place of x to see that Ai(x + γ 1 ) ≤ Ai(a ′ 1 ) ≤ 2Ai(rx + γ 1 ). Finally, suppose rx ≥ a ′ 1 − γ 1 . Because the Airy function is decreasing on (a ′ 1 , ∞), we have Ai(x + γ 1 ) ≤ Ai(rx + γ 1 ), which implies the conclusion of the lemma.
We now check that, at time 0, the value of Z n,A (0), or equivalently Z * n,A (0), cannot be much larger than that of Z n (0). ZAlem Lemma 7.2. Fix ε > 0, and suppose that (1.6) and (1.7) hold. Let A < 0, and define Then P(G n ) > 1 − 3ε for sufficiently large n.
Proof. Since (1.7) holds, it suffices to show that for large n, It follows that In view of (1.7), it follows that for sufficiently large n, Therefore, z n,A (x) ≤ 2z n,0 (x) by Lemma 7.1. It follows from (1.6) that for sufficiently large n, The result follows from (7.2) and (7.3).
Lemma 7.2 tells us that we may restrict our attention to the case in which the initial configuration of particles is such that G n occurs.

Proof.
Recall that H A (t) = L A − βt 2 /9, and that a i,n (s, t) is the position of the ancestor at time s of the ith particle at time t. We divide into three subsets the particles i that are below L A at time t: Likewise, it follows from (5.4) that Finally, noting that H A (t) was chosen so that (2.20) holds when H A (t) < x < L A and H A (t) < y < L A , equation (2.17) implies that for sufficiently large n, Now the result follows from (7.4), (7.5), and (7.6).

A moving barrier barrier_sec
To prove the upper bounds in Proposition 2.3, we need to upgrade the result on Y * in Corollary 7.4 to results about Y and Z. To do so, we need to bound how many particles go above L A . As mentioned earlier, a fixed barrier does not give us accurate enough bounds, so we now define a moving barrier. For A ∈ R and s ≥ 0, let Throughout this argument we will take A < 0, and therefore ∆ A and all its derivatives are nonnegative. We would like to study the process when particles are killed as soon as they hit the curve (Λ A (s)) s≥0 . Letr A x (u, v) be the expected number of particles that hit the curve between times u and v in this modified process, when starting from a single particle at x. rAabound Lemma 7.5. Suppose that A < 0. Then for any t ≥ 0 and x ≤ L A , Proof. Recall that (B t ) t≥0 is a one-dimensional Brownian motion started at x under P x , and T K = inf{t ≥ 0 : B t ≥ K}. Let (G t ) t≥0 be the natural filtration for this Brownian motion. Define Then by the many-to-one lemma, Define a new probability measure Q x by setting where the second expression follows from the first by (stochastic) integration by parts. Combining (7.7) and (7.8)

tells us thatr
Note that since B s ≤ Λ A (s) for all s ≤ T * and ∆ ′ A (s) = Λ ′ A (s) for all s, by the standard integration by parts formula, and so, since x ≤ L A and ∆ ′ A (0) ≥ 0, Under Q x , the process (B t − ∆ A (t)) t≥0 is a Brownian motion started from x, so from above, We now note that for any time t ≥ 0,

Recalling thatr L
and integrating by parts completes the proof of the lemma.

Proof of the upper bounds in Proposition 2.3
hitL Lemma 7.6. Fix ε > 0 and suppose that (1.6) and (1.7) hold and that t ≤ ρ/εβ. Recall the definition of ∆ from (1.4), and let C 1 be the constant from Lemma 2.14. Then there exists a negative real number A ′ , depending on δ, ε, and ∆, such that if A ≤ A ′ , then the probability, conditional on G n , that some particle hits the barrier Λ A (s) at some time s ≤ t and has descendants that survive for an additional time 2C 1 ρ −2 is bounded above by ε for large n.
Proof. First consider a fixed barrier at L A , and let R t be the number of particles killed at this barrier before time t ≥ 0. From Lemma 2.13, Now let R * t be the number of particles that are killed at the moving barrier Λ A (s) for some s ≤ t. By Lemma 7.5 and the bound above, we have Without loss of generality we may assume that ε ≤ 1. Since A < 0, it follows that Recall that It follows that A ′ can be chosen so that if A ≤ A ′ , then E[R * t |F 0 ] ≤ ε∆/2ρ 2 . Because t ≤ ρ/εβ, for sufficiently large n we have ρ 2 /2 ≤ βΛ A (s) ≤ ρ 2 for all s ≤ t. From Lemma 2.14, it follows that conditional on G n , the probability that some particle reaches the boundary before an arbitrary time t and has descendants that survive for an additional time 2C 1 ρ −2 is bounded above by a constant multiple of ε, which is sufficient to imply the result because ε > 0 was arbitrary.
We now have the ingredients to complete the proof of the upper bound Proposition 2.3, in the form of the following lemma. Note in particular that, in view of (1.2), the result (7.9) is stronger than the required conclusion that ρ −2 e −ρL Y n (t n ) → p 0. mainupper Lemma 7.7. Fix ε > 0 and suppose that (1.6) and (1.7) hold, and that the sequence of times t n satisfy βt n /ρ → τ ∈ (0, ∞). Then there exists η > 0 such that P Y n (t n ) ≤ 1 η · β 1/3 ρ 3 e ρL > 1 − 6ε (7.9) mainYupper and P Z n (t n ) ≤ 1 η · β 1/3 ρ 3 e ρL > 1 − 6ε. (7.10) mainZupper Proof. Choose A ′ as in Lemma 7.6, and fix A ≤ A ′ . Fix A * such that for all large n, Recall that, by Corollary 7.4, there exists η > 0 such that on G n , We therefore need to consider those particles that contribute to Y n (t n ) − Y * n,A * (t n ); such particles must be above level L A * at some time before t n .
Note that Λ A (s) ≤ L A * for all s ≤ t n − 2C 1 /ρ 2 . Therefore in order to contribute to Y n (t n ) − Y * n,A * (t n ), a particle must do one of the following: (a) start above L A and survive until time t n ; (b) hit Λ A (s) for some s ≤ t n − 2C 1 /ρ 2 and then survive for an additional time of 2C 1 /ρ 2 ; (c) hit L A * between times t n − 2C 1 /ρ 2 and t n .
By Lemma 5.1, the probability that any particle does (a) tends to 0 as n → ∞. Lemma 7.6 tells us that on G n , the probability that any particle does (b) is bounded above by ε. It therefore remains to consider case (c). Let R ′ be the number of particles that hit L A * between times t n − 2C 1 /ρ 2 and t n . Then on G n , by Lemma 2.13, ρ 2 e −ρL A * β 2/3 e |A * |τ · β 1/3 ρ 3 e ρL .
We have τ ≍ 1 and A * does not depend on n. Also, by (1.2) we have e −τ 3 ρ/10β ≪ β 1/3 /ρ and β/ρ 5 ≪ β 1/3 /ρ 3 . Therefore By Lemma 4.3, the expected contribution from each of these particles to Y n (t n ) is at most e ρL A * . The conditional Markov's inequality gives that for large n, on G n , Since P(G n ) > 1 − 3ε for large n by Lemma 7.2, this completes the proof for Y n (t n ). Because |Ai(x)| ≤ 1 for all x (see table 9.9.1 in [16]), the result for Z n (t n ) follows immediately.

Proof of the lower bound in Proposition 2.3
To prove Proposition 2.3, it remains to establish the lower bound for Z n (t n ). This requires using a second moment argument to control the fluctuations. To do this, we will construct another modification of the original process. We would essentially like to use the moving barrier from Section 7.2, with A chosen positive so that the barrier moves closer to the origin as time increases.
However, our second moment bound Lemma 2.10 holds only for a fixed barrier at L A . Developing the required second moment bounds for the moving barrier would require substantial extra work, and it is much more convenient to instead mimic the moving barrier with a series of fixed barriers that move progressively closer to 0. Fix ε > 0 and choose δ > 0 such that (1.6) holds. Suppose that βt n /ρ → τ ∈ (0, ∞). Let C 12 be a positive constant chosen so that (2.23) is a strict inequality for all n if the right-hand side is multiplied by C 12 . Fix a positive number A large enough that 1/A < τ and Let A 0 = A, and let A k = A + k log(4e 2 ) for positive integers k. Choose J n ∈ N and times for all k ∈ {0, 1, . . . , J n−1 }, which is possible for sufficiently large n because 1/A < τ . Note that Since u Jn = t n ∼ ρτ /β, it follows that for large n, and therefore Choose positive numbers C 13 and C 14 such that 0 < C 13 < C 14 < ∞ and 2 1/3 C 14 2 1/3 C 13 Ai(z + γ 1 ) 2 dz > (Ai ′ (γ 1 )) 2 2 , (7.14) c1c2 which is possible by (2.15) with j = k = 1. Recalling (2.21), we consider a modified process in which particles are killed at time 0 unless they lie in the interval For k ∈ {0, 1, . . . , J n − 1}, particles are killed if they reach L A k between times u k and u k+1 , and then particles are also killed at time u k+1 unless they are in the interval Letting G ′ i (t) be the event that the ith particle at time t in the original process has not been killed by time t in this modification, we define ZAlem2 Lemma 7.8. Fix ε > 0 and suppose that (1.6) and (1.7) hold. Then for sufficiently large n, Thus In view of (1.7), it follows for sufficiently large n that Likewise, by the reasoning that led to (5.15), Therefore, in view of (1.6), Now suppose x < L − 2A/ρ. Then 1 2 (L − x) ≤ L A − x ≤ L − x. Therefore, by Lemma 7.1, we have z n,0 (x) ≤ 2z n,A (x). In view of (7.17), it follows that which implies the result. Lemma 7.9 below gives the lower bound on Z n (t n ) that is needed to complete the proof of Proposition 2.3. In particular, Proposition 2.3 follows directly from Lemmas 7.7 and 7.9. mainZlow Lemma 7.9. Fix ε > 0 and suppose that (1.6) and (1.7) hold. Suppose that βt n /ρ → τ ∈ (0, ∞).
For k ∈ {0, 1, . . . , J n − 1}, we consider the evolution of the process between times u k and u k+1 . Recall that all particles at time u k are in the interval I k , while particles will be killed at time u k+1 unless they are in the interval I k+1 . Particles will also be killed if they reach L A k ,n between these two times. Note that these intervals have been chosen in such a way that if x ∈ I k and y ∈ I k+1 , then the density p L A k u k+1 −u k (x, y) can be approximated using Lemma 2.5 because the error term in (2.17) tends to zero uniformly over x ∈ I k and y ∈ I k+1 as n → ∞. Likewise, Lemma 2.10 can be applied for second moment calculations because any x ∈ I k and y ∈ I k+1 will satisfy the conditions of Lemma 2.10 if n is large enough.
Using (7.11) and recalling that A k = A + k log(4e 2 ), we also have .
It follows that lim sup where the last inequality is from (7.13). Combining this result with (7.18), we get lim sup For all k ∈ {1, . . . , J n } and all positive real numbers a 1 and a 2 , we have lim n→∞ inf y∈I k z n,a 1 (y) z n,a 2 (y) = lim n→∞ sup y∈I k z n,a 1 (y) z n,a 2 (y) = 1. (7.20) ZA1A2 It follows from (7.19) and (7.20 Therefore, by the definition of the events G k,n , and using that P(G 0,n ) > 1 − 3ε for sufficiently large n, we have That is, The result now follows from another application of (7.20).
alphaintlem Lemma 8.3. For any fixed k > 0, we have Proof. Recalling (2.2), making the substitution z = (2β) 1/3 (L A − y), and then using (3.1) and the continuity of the Airy function, we get as claimed.

Proof of Lemma 2.10
Recall that Standard second moment calculations, which go back to early work on branching Markov processes by Ikeda,Nagasawa,and Watanabe (see p. 146 of [22]) give For z ≤ L A , the birth rate b n (z) is bounded by (1.4). Using also that ϕ is bounded and equals zero except on [K A (t), L A ], we have We now split the last term into six parts. We define and note that we may, and will, assume that ρ −2 ≤ t/2 because of (1.2) and the assumption that t β −2/3 . Recall from (2.21) that l(t) = βt 2 /33 and K A (t) = L A − l(t)/2. We write e ρy p L A t−s (z, y) dy 2 dz ds, e ρy p L A t−s (z, y) dy 2 dz ds, e ρy p L A t−s (z, y) dy 2 dz ds, The next seven lemmas, which bound these six terms as well as the first term on the right-hand side of (8.6), will imply Lemma 2.10. Lemma 8.4. Under the assumptions of Lemma 2.10, we have Proof. When K A (t) < x < L A and K A (t) < y < L A , equation (2.18) holds because β −2/3 t. Therefore, by Lemma 2. 5 and (3.6), which implies the lemma because ρ −2 β 2/3 t → ∞ by (1.3) and the assumption that t β −2/3 . LemI Lemma 8.5. Under the assumptions of Lemma 2.10, we have Proof. We use Lemma 2.6 to bound p L A s (x, z) and Lemma 2.
Therefore, using that β −2/3 t, the expression is bounded above by a negative constant, so we can apply (2.19). We get Note that βL A s and ρ 2 s/2 are both bounded above by constants. Using also that A ≥ 0, we get Now applying Lemma 8.3 with k = 1 gives Next, we reverse the roles of z and L A − z and use that α(z) β 1/3 z by (3.5) and (3.6) to get To evaluate the inner integral, we apply Lemma 8.1 with k = 2, t = 2s, and a = L A − x. We now split the argument into two cases depending on the value of x. First, suppose L A − x ≥ ρ −1 . Then, because s ≤ ρ −2 , we have L A − x ≥ ρs, so we can apply (8.1). Noting also that in this case we have ( and therefore Next, suppose instead that L A − x < ρ −1 . This time, we must use (8.1) when s ≤ (L A − x)/ρ and (8.2) when s > (L A − x)/ρ to get The result follows from (8.7), (8.8), and (8.9 Proof. We use Lemma 2.7 to bound p L A s (x, z) and Lemma 2.5 to bound p L A t−s (z, y). Recall from the proof of Lemma 8.5 that when z ≥ L A − l(t), y ≥ K A (t), and s ≤ t/2, we can apply (2.19) to get It follows from (2.1) that when s ≤ 8β −2/3 , the quantity βL A s − ρ 2 s/2 is bounded above by a positive constant. Using also that A ≥ 0, we get Next, we apply Lemma 8.3 with k = 1, interchange the roles of z and L A − z, and use that α(z) β 1/3 z to get z 3 e −ρz e −(z−(L A −x)) 2 /2s dz ds. (8.11) prelim2 To evaluate the double integral, we will apply Lemma 8.1 with k = 3, t = 2s, and a = L − x. This will involve considering two cases, depending on the value of x. First, suppose L A − x ≤ 1/ρ. Then, when s > ρ −2 , we have L A − x < ρs. Therefore, we apply (8.2), discarding the e −a 2 /t term there, to get Combining this with (8.11) and using that L A − x ≤ 1/ρ, we get that (8.10) holds in this case.
Next, suppose L A − x > 1/ρ. We split the double integral in (8.11) into three pieces, denoted J 1 , J 2 , and J 3 , depending on whether s 3/2 · s 1/2 (L A − x − ρs) 3 + s 3/2 e ρ 2 s/2−ρ(L A −x) ds. Now using the bound 1/s ≤ ρ 2 and then making the substitution u = ((L A − x)/ρ − s)ρ 2 /2, so that ds/du = −2/ρ 2 , we get When s > (L A − x)/ρ, we instead apply (8.2) and get Also, using that ρs − (L A − x) ≍ ρs when s ≥ 2(L A − x)/ρ, we have Therefore, by Lemma 4.1, . (8.14) J3bound It follows from (8.12), (8.13), and (8.14) that which, in combination with (8.11), implies that (8.10) also holds when L A − x > 1/ρ. Proof. We may, and will, assume that 8β −2/3 < t/2, as otherwise the term III is zero. Write Note that if s ≥ m, then L A − x ≤ βs 2 /64. Therefore, if s ≥ m and L A − z ≤ βs 2 /64, then so we can use Lemma 2.5 to estimate p L A s (x, z). We can also use Lemma 2.5 to estimate p L A t−s (z, y) as in the proofs of Lemmas 8.5 and 8.6. Therefore, using that A ≥ 0 along with Lemma 8.3 and the bound α(z) β 1/3 z, we get Next, we consider the case in which s ≥ m but L A − z > βs 2 /64. Define In this case, we use Lemma 2.7 to bound p L A s (x, z) and Lemma 2.5 to bound p L A t−s (z, y). Using also Lemma 8.3, we have Because A ≥ 0, we have e βL A s−ρ 2 s/2 ≤ e −2 −1/3 γ 1 β 2/3 s .
We again use Lemma 2.7 to bound p L A s (x, z) and Lemma 2.5 to bound p L A t−s (z, y). Using also Lemma 8.3 and the bound α(z) β 1/3 z, we get We now estimate the inner integral using Lemma 8.1 with k = 3, t = 2s, and a = L − x. We need to consider three cases. First, suppose s ≥ 2(L A − x)/ρ. Then L A − x ≤ 1 2 ρs, so we use (8.2) and the fact that s ≤ m to get Combining (8.25) with (8.26), and using (8.19) again along with the fact that m = 8 (L A − x)/β, we get ρs, so again we use (8.2). This time, we keep the s 2 term in the denominator, and we get Combining this result with (8.25) and (8.19), we get We may assume that (L A − x)/ρ > 8β −2/3 , which implies that ρ(L A − x) ≥ 8ρ 2 β −2/3 → ∞ and therefore s 3/2 ≤ (L A − x) 3/2 ρ −3/2 ≪ (L A − x) 3 . It follows that ∞ 0 e −ρz z 3 e −(z−(L A −x)) 2 /2s dz s 1/2 (L A − x) 3 e ρ 2 s/2 e −ρ(L A −x) . Therefore, Because L A ≤ 2ρ 2 /3β for sufficiently large n, we have e βL A (L A −x)/ρ ≤ e 2ρ(L A −x)/3 , and therefore Proof. We may, and will, assume that 8β −2/3 < t/2, as otherwise the term IV is zero. When s ≥ t/2, x ≥ K A (t) = L A − l(t)/2, and z ≥ L A − l(t), we have Therefore, using that β −2/3 s, we see that is bounded above by a negative constant, so we can apply Lemma 2.5 to approximate p L A s (z, y) and use (2.19).
Finally, using the assumption that t β −2/3 , we have Now, we consider the remaining case in which L A −z ≤ β(t−s) 2 /128 and L A −y > β(t−s) 2 /64. Define e ρy p L A t−s (z, y) dy 2 dz ds.
Proof. As in the proof of Lemma 8.8, we can apply Lemma 2.5 to estimate p L A s (z, y). To bound p t−s (z, y), we will use either the bound from Lemma 2.6 or the bound from Lemma 2.7, whichever is smaller. Using also that, when t − s ≤ β −2/3 , we have t − s e −(z−y) 2 /2(t−s) dy 2 dz ds.
Interchanging the roles of s and t − s, z and L A − z, and y and L A − y, we get V β 1/3 z A (x)e ρL Now noting that z 2 /s ≥ z/ √ s whenever either of these expressions is larger than one, it follows that l(t)/2 0 min 1, yz s · 1 √ s e −(z−y) 2 /2s dy ≤ min 1, z 2 s .
Proof. We use Lemma 2.7 to bound both p L A s (x, z) and p L A t−s (z, y) and get (L A − y)e −(y−z) 2 /2(t−s) dy 2 dz ds.