From $1$ to $6$: a finer analysis of perturbed branching Brownian motion

The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes $\sigma_1^2=\sigma_2^2=1$ which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing $\sigma_1^2=1\pm t^{-\alpha}$ and $\sigma_2^2=1\pm t^{-\alpha}$. We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the iid case $\frac{1}{2\sqrt 2}\ln(t),\;\frac{3}{2\sqrt 2}\ln(t)$ and $\frac{6}{2\sqrt 2}\ln(t)$ when $0<\alpha<\frac{1}{2}$. This is due to the localisation of extremal particles at the time of speed change which depends on $\alpha$ and differs from the one in standard branching Brownian motion. We also establish in all cases the asymptotic law of the maximum and characterise the extremal process, which turns out to coincide essentially with that of standard branching Brownian motion.


Introduction
So-called log-correlated (Gaussian) processes have received considerable attention over the last few years; see, e.g., [2,4,8,9,27]. One of the reasons for this is that they represent processes where the correlations are on the borderline of becoming relevant for the properties of the extremes of the process. A paradigmatic example for such processes is branching Brownian motion (BBM) [1,33]. This process has been intensely investigated from the point of view of extreme value theory over the last 40 years; see, e.g., [2, 5-7, 10, 15, 17-19, 30]. To understand what we mean by BBM being borderline, it is useful to consider BBM as a special A.0/ h 0 and A.1/ h 1that is increasing and right-continuous. Given such a function, so-called variable speed branching Brownian motion [11,12,20,21,31] can then be constructed in two equivalent ways. 1 Fix a time horizon t and let  Branching Brownian motion with speed function 2 t is constructed like ordinary branching Brownian motion except that, if a particle splits at some time s < t, then the offspring particles perform variable speed Brownian motion with speed function 2 t ; i.e., their laws are independent copies of fB r B s g t!r!s , all starting at the position of the parent particle at time s. We assume here and throughout this paper that particles in BBM branch after an exponential time of parameter one with probability p k into k independent copies of themselves where the branching law p k satisfies I ih1 p k h 1, I kh1 kp k h 2, and K h I kh1 k.k 1/p k < I.
This ensures, in particular, that the process cannot die out. It also normalizes that number of particles at time t, n.t/ to satisfy En.t/ h e t .
Alternatively, variable speed BBM can be constructed as a Gaussian process indexed by a continuous-time Galton-Watson tree with mean zero and covariances (1.3) Ex k .s/x`.r/ h 2 t .d.x k .t/; x`.t// s r/; where the x k label the n.t/ particles present at time t and d.x k .t/; x`.t// is the time of the most recent common ancestor of the particles labeled k and`in the Galton-Watson tree.
The authors' interest in this model was actually sparked by the second construction, which exhibits the connection to the generalized random energy models (GREM) introduced by Derrida [23] (see also [20]). These models were introduced as toy models for spin glasses for which the structure of extreme values is important. The major goal here is to understand the dependence of the structure of extremes on the covariance function. An analysis of the order of the maximum was carried out in [13,14]. Already in this work, the phase transition happening at the identity function (which is described in more detail below) is visible. This is a main motivation for the study of arbitrary covariance functions and in particular this work, as it sheds light on how this transition exactly happens on a microscopic level.
After this small detour, let us now connect the two definitions of branching Brownian motion. The case A.x/ h x corresponds to standard branching Brownian motion. The behavior of the extremes of these processes are dramatically different according to whether A stays below x or whether it crosses this line: (i) if A.x/ < x for all x P .0; 1/, then, to first sub-leading order, ( A is a piecewise linear function [11,12].. In this paper we have a closer look at the apparent discontinuities that happen when A crosses the identity line (see (1.8)). To do so, we consider functions A h A t that depend explicitly on the time horizon t. Kistler and Schmidt [28] have considered the case when A t is a step function with step sizes t 1 and step heights t 1 that converges to A.x/ h x from below. They showed that in this case, the logarithmic correction is given by 3 2 2 p 2 ln t, which interpolates nicely between cases (i) and (ii).
Here we consider piecewise linear functions that lie slightly above or below Interpreting this result in the context of the F-KPP equation, this hints at a continuity result for the speed of the front positions.

Localization
The key observation that will be needed to prove this and more detailed facts is a localization result on the position of the ancestors of extremal particles a time t=2. It is known that in the case when 2 1 h 1gO.1/ the ancestors of extremal particles at time t are also extremal at time t=2, and so are just a logarithm of t below p 2 t 1 . For standard BBM, these particles are O. p t/ below p 2 t=2. In the case 2 1 h 1 O.1/, these particles are even further below, namely by p 2. 1 2 1 /t=2 [11]. We will show (in Chapters 3 and 4, resp.), that the ancestors of extremal particles at time t are below p 2 1 t=2 by O.t / in the case 2 1 h 1 g t , and by p 2 t 1 =4gO. p t/ in the case 2 1 h 1 t , when P .0; 1=2 (see Figure 1.1).
Aficionados of BBM will readily infer (1.12) and (1.13) from this information. To actually prove this is, however, a bit more delicate. The basic strategy is similar to that used in the case of two-speed BBM with 2 1 < 1 in [11], but there are some interesting twists.
2 Note that p 2 1 g 2 2 t % p 2.t t 1 2 =2/, which is already different from the BBM case if

1=2.
Apart from the analysis of the log-correction to the value of the maximum, we also analyze the law of the maximum and the nature of the extremal process in these cases. Of course, in both cases the law of the maximum converges to a randomly shifted Gumbel distribution. Less obviously, whenever P .0; 1=2/, the random shift is always given by the derivative martingale (see (1.16) below). The extremal process has the same structure as in BBM, i.e., a decorated Cox process, where the decoration process is independent of .
In the remainder of this paper, when we consider the case 1 > 2 , we always set 2 1 h 1 g t ; 2 2 h 1 t In both cases, this is correct for 0 < 1=2. If > 1=2, all is exactly as in standard BBM.
We will denote particles of two-speed BBM with variances 2 1 on 0; t=2 and 2 2 on t=2; t by z x k .s/ and those of standard BBM by x k .s/.
Before stating the main result of this paper, let us recall the two key martingales that were introduced by Lalley [11]).
We can now state the main results of this paper. where Z is the limit of the derivative martingale and C./ is the positive constant shift, always the same as that of standard BBM (see [7]), if > 0.

Outline of the Paper
The remainder of this paper is organized as follows. In Section 2 we recall some facts on the tail behavior of solutions of the F-KPP equation that form the crucial input in the analysis. Sections 3 and 4 contain the proof of Theorem 1.1. We deal separately with the cases 1 > 1 and 1 < 1. The structure of the proof is the same in both cases, but the details of the calculations are different and it appears easier to follow the arguments in each case rather then to jump back and forth.
The way both chapters are organized is as follows. First, we show where the extremal particles are localized at the change time t=2. Then we exploit the branching 2 ), everything is as in standard BBM. In the northwest regime, the order of the maximum and the extremal process are a concatenation of two such processes for standard BBM. In the regime in between (0 < < 1 2 ), the order of the maximum interpolates smoothly between the surrounding regimes. In the southeast regime, the order of the maximum coincides with the one in the i.i.d. case. The extremal process is similar to the one for BBM but the martingale appearing is different. In the regime, with 2 1 h 1 t , 0 < < 1 2 , the order of the maximum interpolates smoothly between the i.i.d. and the BBM order of the maximum. Observe that in the three middle regimes the extremal process coincides up to constant shift with the one of standard BBM and the martingale is always the derivative martingale.
property at time t=2 to set up a recursion where the tail asymptotics of the law of the maximum of the BBMs after time t=2 are used. This results in a formula that is already somewhat reminiscent of the Lalley-Sellke representation [30] of the limiting distribution of the maximum of BBM. However, to prove convergence, we need to exhibit more independence by splitting paths at time t for some suitable small . This results in an expression that in all cases involves a slight modification of the derivative martingale that we then show to converge towards the limit of the usual derivative martingale.
In Section 5 we prove convergence of the the Laplace functionals and hence the extremal process. This is essentially identical to the proof of the law of the maximum and requires just a slight extension of the results on the asymptotics of solutions of the F-KPP equation to the case of weakly t-dependent initial conditions.

Preliminaries about BBM
In this section we collect some known results about standard branching Brownian motion. A fundamental property of BBM is its relation to the Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP) equation [22,29] that was established by Ikeda, Nagasawa, and Watanabe [24][25][26] and McKean [32]. Namely, if we set, for some function f 0; 1 3 0; 1, The following proposition is based on the deep analysis of the behavior of solutions to the F-KPP equation presented in Bramson's monograph [16]. PROOF. The proof of this proposition is a direct adaptation of the proofs of the corresponding propositions in [7,11] for the cases x $ p t and x $ t.
Remark 2.2. Choosing for f the Heaviside function, this proposition implies in particular that, for x > 0, The rougher bound below follows by using the many-to-one lemma and standard Gaussian asymptotics: LEMMA 2.3. For any x P R g , 3 The Law of the Maximum: The Case 2 1 h 1 g t The aim of this section is to prove Theorem 1.1 in the case when 2 i.e., to show that where we set C 2 3=2 C . In this section we will always write m.t/ m g .t/. Z is the limit of the derivative martingale and C is the

Localization of Paths
To prove (3.1), we need to control the position of particles until time t=2. To this end, we define three sets on the space of paths X R g 3 R. The first controls the position at time s. The second ensures that the path of the particle does not exceed a certain value, and the third controls the positions of particles at time t .
In the case of standard BBM, it was shown in Bramson [15] (see also the detailed analysis in [5]) that the positions of particles that are near the maximum at time t are at time t=2 in a window of order p t below p 2 t=2. In the case of 2-speed BBM with 1 < 2 , it was shown in [11] that the corresponding window is of width p t around p 2 2 1 t=2, which is a linear order in t below the level of the maximal particles at time t=2 (which is near p 2 1 t=2). If 1 > 1, then extremal particles descend from the actual extremal particles at time t=2. So we expect that in our case, we see a transition from p t to O.1/ as we vary .
2 h 1 t . For any > 0, there is r 0 < I such that for all r > r 0 and for all t large enough, PROOF. The event considered depends only on standard BBM up to time t=2.
The well-known estimate for standard BBM follows from Bramson's results in [15], see also [5]. We have to distinguish the cases where´ K p t and the rest. In the former, we can proceed as in the case above and we get, up to vanishing terms, for any K > 0, ( which tends to zero rapidly as t 4 I. This concludes the proof.
The next proposition states that H holds for all extremal particles for 0 < < 1=2. This is a weaker form of the localization results shown in [5]. PROOF. To prove this proposition, we may use Proposition 3.2 and the fact that any path starting at zero, ending at some p 2 t=2 ´with´P At ; Bt , and staying below the line p 2 s will not be above p 2 t t at time t with high probability.
To do so, we decompose a bridge in time t=2 from 0 to´into two pieces, one from 0 to p 2 t y in time t and one from p 2 t y to p 2 t=2 ´in time t £ t=2 t . Then the probability that the first bridge stays below p 2 s is, to leading order in t, given by while the probability for the second bridge is 2y´=t £ . These estimates follow from lemma 2.2 in [16]. Thus the probability that the bridge is above p 2 t t is given by The right-hand side tends to zero for any < 1=2, which implies the assertion of the proposition.
The following simple lemma shows that if a condition holds for all paths that exceed some level, then this condition can also be imposed on the paths when computing the probability that the maximum stays below that level. LEMMA 3.4. Let x k ; k h 1; : : : ; n be path-valued random variables and G be any event such that, for some > 0, (3.20) P W k n fx k .t/ > yg fx k P G g ¡ ! P W k n x k .t/ > y ¡ : As the term e x appears in the conditional expectation with respect to F t , we need to compute E.Y k .t/jF t / and E.Z k .t/jF t /, and control E.Y k .t/ 2 jF t /, E.Z k .t/ 2 jF t /, and E.Y k .t/Z k .t/jF t /.

Computation of the Main Term
We begin with the computation of the averages of the McKean and the derivative martingale terms.  This concludes the proof of the lemma.
Remark 3.6. It is curious to see that the terms p 2t x k .t / appear and recreate the derivative martingale as a factor of EZ k .t/. If we had been a bit more sloppy and used as the probability for the bridge just t =2 t t £ , we would instead have gotten just a factor t 1=2 multiplying the McKean martingale. But nothing would have changed, since by a result of Aïdékon and Shi [3], this would converge in probability to a limit that has the same law as the limit of the derivative martingale.

Controlling the Second Moment
We now show that the expectations of the quadratic terms are bounded by a polynomial term in t ; see (3. To the level of precision we care about, the integral in the square can be bounded by the maximum of its integrand, i.e., The remaining integral over w is trivially bounded by p s=.2t £ s/, which is smaller than 1, and we are done.

Towards the Derivative Martingale
We have seen that  The next lemma asserts that this is indeed the case.

Control of the Almost Martingale
LEMMA 3.7. With the notation above, p 2 t x k .t // 3 Z; t 4 I: Next, we introduce, for some 1 > > 1=2, (3.64) 1 h 1 x k .t /> p 2 t t g 1 x k .t / p 2 t t : We control the two resulting terms separately and start with (3.65) n.

Conclusion of the Proof
Using Lemma 3.7 we see that indeed the right-hand side of (3.59) converges, as first t 4 I and then A 5 0 and B 4 I, in probability to (3.76) C Z e p 2y p 2 : Together with the fact that the term in (3.58) converges to 0, we get that (3.34) converges to (3. Z is the limit of the derivative martingale and C is a positive constant.
The structure of the proof is identical to that in the previous section.

Localization of Paths
To prove Theorem 4.1 we need to control the position of particles until time t=2. Only the position of the path at time t=2 needs to be modified from the previous section. Therefore we redefine  The terms in the last exponential can be written as (4.14) . 2  For any finite r, this tends to 0 as A 4 I. This concludes the proof of the proposition.

Recursive Structure
As in the previous section, and with the same notation, we write  Here we used that t 1 t =2 h t . Note that this time there is no term involving Z k ! As in the case 1 > 2 , we can effectively replace in (4.30) exp.¡/ by 1 g .¡/, compute the conditional expectation, and return then to exp.¡/. This gives  with L P N, c`> 0, and u`P R (see [10,11] As in the previous chapters, we would like to interpret the conditional expectation in the product as a solution of the F-KPP equation and use the asymptotics of these solutions. However, there is a small problem due to the fact that the 2 that multiplies x k j .t=2/ depends on t. We will see that this problem can be solved rather easily with the help of the maximum principle.
To see this, consider, for fixed t P R and f R 3 R g , where f t .
x/ h f .x 2 .t//. Then, for fixed t, 1 v t is a solution of the F-KPP equation with initial condition 1 v t .0; x/ h 1 f t .x/. Provided that f (and f t ) satisfies the assumptions of Bramson's theorem, we can derive the large-s asymptotics for v t . However, we want to look at the asymptotics when s h t=2 and t 4 I. Since in our cases, f t .
x/ 3 f .x/ as t 4 I, the initial conditions satisfy Bramson's conditions uniformly in t and bounds on v t .s; x/ for large s hold uniformly in t.
Fortunately, the maximum principle allows us to overcome this difficulty.
LEMMA 5.1. Assume that f t is such that, for all t > t 0 and all x ! 0, (5.3) f .x/ f t .x/ f t 0 .x/: Then, for all x ! 0 and all t > t 0 ,