Fisher-KPP equation with small data and the extremal process of branching Brownian motion

The standard binary branching Brownian motion (BBM) on ${\mathbb{R}}$ is a process that starts at the initial time $t=0$ with one particle at the position $x=0$. The particle performs a Brownian motion until a random, exponentially distributed with parameter $1$, time $\tau>0$ when it splits into two off-spring particles. Each of the new particles starts an independent Brownian motion at the branching point. They have independent, again exponentially distributed with parameter $1$, clocks attached to them that ring at their respective branching times. At the branching time, the particle that is branching produces two independent off-spring, and the process continues, so that at a time $t>0$ we have a random number $N_{t}$ of Brownian particles. It will be convenient for us to assume that the variance of each individual Brownian motion is not $t$ but $2t$.

A remarkable observation by McKean [19] is a connection between the location of the maximal particle for the BBM and the Fisher-KPP equation

$$\dfrac{\partial{u}}{\partial{t}}=\frac{\partial^{2}u}{\partial x^{2}}+u-u^{2},$$

(1.1)

with the initial condition $u(0,x)=\mathbbm{1}(x<0)$. Let $x_{1}(t)\geq x_{2}(t)\geq\ldots\geq x_{N_{t}}(t)$ be the positions of the BBM particles at time $t$. McKean has discovered an exact formula

$$u(t,x)=\mathbb{P}[x_{1}(t)>x].$$

(1.2)

More generally, given a sufficiently regular function $g(x)$, the solution to the Fisher-KPP equation (1.1) with the initial condition $u(0,x)=g(x)$ can be written as

	$\displaystyle u(t,x)$	$\displaystyle=1-{\mathbb{E}}_{x}\Big{[}\prod_{j=1}^{N_{t}}(1-g(x_{j}(t))\Big{]}=1-{\mathbb{E}}_{0}\Big{[}\prod_{j=1}^{N_{t}}(1-g(x+x_{j}(t))\Big{]}$		(1.3)
		$\displaystyle=1-{\mathbb{E}}_{0}\Big{[}\prod_{j=1}^{N_{t}}(1-g(x-x_{j}(t))\Big{]}=1-{\mathbb{E}}_{-x}\Big{[}\prod_{j=1}^{N_{t}}(1-g(-x_{j}(t))\Big{]},$

so that (1.1) is a special case of (1.3) with $g(x)=\mathbbm{1}(x<0)$. Here, ${\mathbb{E}}_{x}$ refers to a BBM that starts at $t=0$ with a single particle at the position $x\in{\mathbb{R}}$ and not at $x=0$. The slightly unusual form of (1.3) in the right side, with the process starting at $(-x)$, will be convenient when we look at the limiting measure of the BBM, also with a flipped sign, as in (1.10) below.

McKean’s interpretation has been used by Bramson [7, 8] to establish the following result on the long term behavior of the solutions to the Fisher-KPP equation. It has been known since the pioneering work of Fisher [12] and Kolmogorov, Petrovskii and Piskunov [16] that the Fisher-KPP equation admits traveling wave solutions of the form $u(t,x)=U_{c}(x-ct)$ that satisfy

$$-cU_{c}^{\prime}=U_{c}^{\prime\prime}+U_{c}-U_{c}^{2},~{}~{}U_{c}(-\infty)=1,~{}~{}U_{c}(+\infty)=0,$$

(1.4)

for all $c\geq c_{*}=2$. We will denote by $U(x)$ the traveling wave $U_{2}(x)$ that corresponds to the minimal speed $c_{*}=2$:

$$-2U^{\prime}=U^{\prime\prime}+U-U^{2},~{}~{}U(-\infty)=1,~{}~{}U(+\infty)=0.$$

(1.5)

Solutions to (1.5) are defined up to a translation in space. We fix the particular translate by requiring that the traveling wave has the asymptotics

$$U(x)\sim(x+k_{0})e^{-x},\hbox{ as $x\to+\infty$},$$

(1.6)

with the pre-factor in front of the right side equal to one, and some $k_{0}\in{\mathbb{R}}$ that is not, to the best of our knowledge, explicit. Let now $u(t,x)$ be the solution to (1.1) with the initial condition $g(x)$ such that $0\leq g(x)\leq 1$ for all $x\in{\mathbb{R}}$, and $g(x)$ is compactly supported on the right – there exists $L_{0}$ such that $g(x)=0$ for all $x\geq L_{0}$. It was already shown in [16], for the particular example of $g(x)=\mathbbm{1}(x\leq 0)$, that there exists a reference frame $m_{kpp}(t)$ such that $m_{kpp}(t)/t\to 2$ as $t\to+\infty$ and

$$u(t,x+m_{kpp}(t))\to U(x)\hbox{ as $t\to+\infty$},$$

(1.7)

uniformly on semi-infinite intervals of the form $x\geq K$, for each $K\in{\mathbb{R}}$ fixed. Bramson has refined this result, showing that there exists a constant $\hat{s}[g]$ that is known as the Bramson shift corresponding to the initial condition $g$, such that

$$u(t,x+m(t))\to U(x+\hat{s}[g])\hbox{ as $t\to+\infty$},$$

(1.8)

uniformly on semi-infinite intervals of the form $x\geq K$, for each $K\in{\mathbb{R}}$ fixed, with

$$m(t)=2t+\frac{3}{2}\log t.$$

(1.9)

Note that we have chosen the sign of $\hat{s}[g]$ in (1.8) in the way that makes the shift positive for ”small” initial conditions that we will consider later. In that sense, at a time $t\gg 1$, the solution is located at the position $m(t)$. A shorter probabilistic proof of this convergence was given recently in [24], and PDE proofs of various versions on Bramson’s results have been obtained in [14, 18, 22, 25], with further refinements in [13, 15, 23] and especially in the recent fascinating paper [5].

Motivated by the above discussion, one may consider not just the maximal particle but the statistics of the BBM process re-centered at the location $m(t)$, and ask if it can also be connected to the solutions to the Fisher-KPP equation. Let $x_{1}(t)\geq x_{2}(t)\geq\ldots$ be the positions of the BBM particles at time $t$, and consider the BBM measure seen from $m(t)$:

$\displaystyle\mathcal{X}_{t}=\sum_{k\leq N_{t}}\delta_{m(t)-x_{k}(t)}.$

(1.10)

Recall that $N_{t}$ is the number of particles alive at time $t$. It was shown in [1, 2, 3, 10] that there exists a point process $\mathcal{X}$ so that we have

$\displaystyle\mathcal{X}_{t}\Rightarrow\mathcal{X}=\sum_{k}\delta_{\chi_{k}}~{}~{}\hbox{ as $t\to+\infty$},$

(1.11)

with $\chi_{1}\leq\chi_{2}\leq\ldots$, so that $\chi_{1}$ corresponds to the maximal particle in the BBM, $\chi_{2}$ to the second largest, and so on. In what follows we will call $\mathcal{X}$ the limiting extremal process or simply extremal process. Sometimes it is also called in the literature the decorated Poisson point process, see e.g. [1]. The properties of the limit measure $\mathcal{X}$ are closely related to the long time limit of the derivative martingale introduced in [17]:

$$Z_{t}=\sum_{k\leq N_{t}}(2t-x_{k}(t))e^{-(2t-x_{k}(t))}\rightarrow Z~{}\hbox{ as $t\to+\infty$,~{} ${\mathbb{P}}$-a.s.}$$

(1.12)

It turns out that there exists a direct connection between $\mathcal{X}$, $Z$ and the Bramson shift via the Laplace transform of $\mathcal{X}$. As explained in Appendix C of [10], see also [3], the results of [17] imply that for any test function $\psi\geq 0$ that is compactly supported on the right, we have the identity

$${\mathbb{E}}\left[e^{-\mathcal{X}(\psi)}\right]={\mathbb{E}}\Big{[}e^{-Ze^{-\hat{s}[\hat{\psi}]}}\Big{]},~{}~{}\hat{\psi}(x)=1-e^{-\psi(x)}.$$

(1.13)

Here, for a measure $\mu$ and a function $f$ we use the notation

$$\mu(f)=\int_{\mathbb{R}}f(x)\mu(dx).$$

Note that $0\leq\hat{\psi}(x)\leq 1$, and $\hat{\psi}(x)$ is also compactly supported on the right, so that its Bramson shift $\hat{s}[\hat{\psi}]$ is well-defined and finite. In addition to a special case of (1.13) with $\hat{\psi}(x)=\mathbbm{1}(x\leq 0)$, it was shown in [17] that for each $y\in{\mathbb{R}}$ we have

$\displaystyle{\mathbb{E}}\Big{[}e^{-Ze^{-y}}\Big{]}=1-U(y).$

(1.14)

This together with (1.13), characterizes the Laplace transform of $\mathcal{X}$: for any test function $\psi\geq 0$ compactly supported on the right, we have the duality identity

$\displaystyle{\mathbb{E}}\left[e^{-\mathcal{X}(\psi)}\right]=1-U(\hat{s}[{\hat{\psi}}]),~{}~{}\hat{\psi}=1-e^{-\psi}.$

(1.15)

Let us note that the normalization of the traveling wave in (1.6) implies, in particular, that extra shifts appear neither in (1.14), nor in (1.15). A helpful discussion of the normalization constants in these identities can be found in Chapter 2 of [6]. We also explain this in Section 2.

It is important to note that, in fact, the results in [1], [3] and in Appendix C of [10] imply the conditional version of (1.13), see Lemma 2.1 below:

$${\mathbb{E}}\left[e^{-\mathcal{X}(\psi)}|Z\right]=e^{-Ze^{-\hat{s}[\hat{\psi}]}}.$$

(1.16)

In principle, (1.16) completely characterizes the conditional distribution of the measure $\mathcal{X}$ in terms of its conditional Laplace transform. However, the Bramson shift is a very implicit function of the initial condition, and making the direct use of (1.16) is by no means straightforward. One of the goals of the present paper is precisely to make use of this connection to obtain new results about the extremal process $\mathcal{X}$.

Let us first illustrate what kind of results on the Bramson shift we may need on the example of the asymptotic growth of $\mathcal{X}$, Theorem 1.1 in [11], originally conjectured in [10]. This result says that there exists $A_{0}>0$ so that

$$\frac{1}{A_{0}Zxe^{x}}\mathcal{X}((-\infty,x])\to 1,~{}~{}\hbox{ as $x\to+\infty$, in probability.}$$

(1.17)

The proof of (1.17) in [11] uses purely probabilistic tools. In order to relate this result to the Bramson shift and the realm of PDE, we can do the following. Consider the shifted and rescaled version of the measure $\mathcal{X}$, as in (1.17):

$\displaystyle Y_{n}(dx)=n^{-1}e^{-n}\mathcal{X}_{n}(dx),$

(1.18)

where

$\displaystyle\mathcal{X}_{n}=\sum_{k}\delta_{\chi_{k}-n},$

(1.19)

so that

$$\frac{1}{Zne^{n}}\mathcal{X}((-\infty,n])=\frac{1}{Zne^{n}}\mathcal{X}_{n}((-\infty;0])=\frac{1}{Z}Y_{n}((-\infty;0]).$$

(1.20)

We may analyze the conditional on $Z$ Laplace transform of $Y_{n}$ using (1.16): given a non-negative function $\phi_{0}(x)$ compactly supported on the right, we have

$${\mathbb{E}}\left[e^{-Y_{n}(\phi_{0})}|Z\right]={\mathbb{E}}\left[e^{-n^{-1}e^{-n}\mathcal{X}_{n}(\phi_{0})}|Z\right]={\mathbb{E}}\left[e^{-\mathcal{X}(\phi_{n})}|Z\right]=\exp\big{\{}-Ze^{-\hat{s}[\psi_{n}]}\big{\}},$$

(1.21)

with

$$\phi_{n}(x)=n^{-1}e^{-n}\phi_{0}(x-n),~{}~{}~{}\psi_{n}(x)=1-\exp\{-\phi_{n}(x)\}.$$

(1.22)

Note that $\psi_{n}$ is also compactly supported on the right, so that its Bramson shift is well-defined. Furthermore, for $n\gg 1$ the function $\psi_{n}(x)$ is small: it is of the size $O(n^{-1}e^{-n})$, as is $\phi_{n}(x)$. Thus, (1.21) relates the understanding of the conditional on $Z$ weak limit of $Y_{n}$ to the asymptotics of the Bramson shift for small initial conditions for the Fisher-KPP equation (1.1), and this is the strategy we will exploit in this paper to obtain limit theorems for the process $\mathcal{X}$. Let us stress that a connection between the limiting statistics of BBM and the Bramson shift for small initial conditions was already made in [10], though with a slightly different objective in mind, and in a different way.

We now state the results for the Bramson shift of the solutions to the Fisher-KPP equation

$$\dfrac{\partial{u_{\varepsilon}}}{\partial{t}}=\frac{\partial^{2}u_{\varepsilon}}{\partial x^{2}}+u_{\varepsilon}-u_{\varepsilon}^{2},$$

(1.23)

with a small initial condition

$$u_{\varepsilon}(0,x)=\varepsilon\phi_{0}(x),$$

(1.24)

that we will need for studying the limiting behavior of $\mathcal{X}$. Here, $\varepsilon\ll 1$ is a small parameter, and the function $\phi_{0}(x)$ is non-negative, bounded and compactly supported on the right: there exists $L_{0}\in{\mathbb{R}}$ such that $\phi_{0}(x)=0$ for $x\geq L_{0}$. We will use the notation $x_{\varepsilon}=\hat{s}[\varepsilon\phi_{0}]$ for the Bramson shift of $\varepsilon\phi_{0}$:

$$|u_{\varepsilon}(t,x+m(t)|\to U(x+x_{\varepsilon})\to 0\hbox{ as $t\to+\infty$. uniformly on compact intervals in $x$,}$$

(1.25)

We chose the sign of $x_{\varepsilon}$ in (1.25) so that $x_{\varepsilon}>0$ for $\varepsilon>0$ sufficiently small. The following proposition gives the asymptotic behavior for $x_{\varepsilon}$ for small $\varepsilon>0$ that is sufficiently precise to recover (1.17).

To formulate the convergence result, let $\mathcal{C}_{c}$ (resp. $\mathcal{C}^{+}_{c}$) be the space of continuous (resp. non-negative continuous) compactly supported functions on ${\mathbb{R}}$, and $\mathcal{C}_{bc}$ (resp. $\mathcal{C}^{+}_{bc}$) be the space of bounded continuous (resp. non-negative bounded continuous) functions on ${\mathbb{R}}$ compactly supported on the right. Let $\mathcal{M}_{v}$ (resp. $\mathcal{M}_{v}^{+}$) be the space of signed (resp. non-negative) Radon measures on ${\mathbb{R}}$ such that $|\mu|((-\infty,0])<\infty$, equipped with the topology $\tau_{v}$ generated by

$$\mu_{n}\underset{n\rightarrow\infty}{\overset{\tau_{v}}{\longrightarrow}}\mu\iff\mu_{n}(f)\underset{n\rightarrow\infty}{\longrightarrow}\mu(f)\quad\forall f\in{\mathcal{C}}_{bc}.$$

An immediate corollary of Proposition 1.1 is the following version of (1.17). We set

$$\mu(dx)=\frac{1}{\sqrt{4\pi}}e^{x}\,dx,$$

(1.28)

so that

$$\bar{c}=\mu(\phi_{0}).$$

(1.29)

In other words, $Y_{n}(dx)$ looks like an exponential shifted by $\log Z$ to the left. This theorem also explicitly identifies the constant $A_{0}$ in (1.17). Accordingly, we can reformulate Theorem 1.2 as follows: consider the measures $\mathcal{X}^{*}_{n}$ shifted by $\log Z$:

$$\mathcal{X}^{*}_{n}\equiv\sum_{k}\delta_{\chi_{k}+\log Z},$$

(1.31)

and

$$Y^{*}_{n}(dx)=n^{-1}e^{-n}\mathcal{X}^{*}_{n}(dx).$$

(1.32)

This gives the following version of Theorem 1.2:

As we have mentioned, Theorem 1.2 and Corollary 1.3 are not really new, except for identifying the constant $A_{0}$, even though the approach via the asymtptotics of the Bramson shift produces an analytic rather than a probabilistic proof. In order to obtain genuinely new results on the fluctuations of $Y_{n}(dx)$ around $Z\mu(dx)$, we will need a finer asymptotics for the shift $x_{\varepsilon}$ than in Proposition 1.1. Let us define the constant

$\displaystyle\bar{c}_{1}$

$\displaystyle=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}xe^{x}\phi_{0}(x)dx,$

(1.34)

that depends on the initial condition $\phi_{0}$, as does $\bar{c}$ in (1.27), and universal constants

$$g_{\infty}=\int_{0}^{1}e^{z^{2}/4}\int_{z}^{\infty}e^{-y^{2}/4}dydz-2\int_{1}^{\infty}e^{z^{2}/4}\int_{z}^{\infty}\frac{1}{y^{2}}e^{-y^{2}/4}dy,$$

(1.35)

and

$$m_{1}=\frac{3}{2}g_{\infty}+k_{0}+\frac{1}{2},$$

(1.36)

that do not depend on $\phi_{0}$. Here, $k_{0}$ is the constant that appears in the asymptotics (1.6) for $U(x)$. The following theorem allows us to obtain convergence in law of the fluctuations of $Y_{n}$.

The first two terms in (1.26) and (1.37) have been predicted in [10] in addressing a different BBM question, using an informal Tauberian type argument that we were not able to make rigorous. The rest of terms have not been predicted, to the best of our knowledge. The proof in the current paper does not seem to be directly related to the arguments of [10] but the general approach to the statistics of BBM via the Bramson shift asymptotics for small initial conditions comes from [10].

Theorem 1.2 indicates that to obtain the limiting behavior of the fluctuations of $Y_{n}(dx)$ one should consider a rescaling of the signed measure

$$Y_{n}(dx)-Z\mu(dx)=Y_{n}(dx)-\frac{1}{\sqrt{4\pi}}e^{(x+\log Z)}dx.$$

(1.38)

It turns out, however, that there is an extra small deterministic shift of the exponential profile that needs to be performed before the rescaling: a better object to rescale is not as in (1.38) but

$$Y_{n}(dx)-\frac{1}{\sqrt{4\pi}}e^{(x+\log Z+e_{n})}dx,$$

(1.39)

with an extra deterministic correction

$$e_{n}=\log\Big{(}1+\frac{2\log n}{n}\Big{)}\approx\frac{2\log n}{n}.$$

(1.40)

The properly rescaled measures are

		$\displaystyle V_{n}(\phi_{0})=n\Big{(}Y_{n}(\phi_{0})-(1+\frac{2\log n}{n})Z\mu(\phi_{0})\Big{)},$		(1.41)
		$\displaystyle V^{*}_{n}(\phi_{0})=n\Big{(}Y^{*}_{n}(\phi_{0})-(1+\frac{2\log n}{n})\mu(\phi_{0})\Big{)}.$

Let us set

$$\nu(dx)=(4\pi)^{-1/2}xe^{x}\,dx,$$

(1.42)

and let $\{R_{t},t\geq 0\}$ be a spectrally positive $1$-stable stochastic process with the Laplace transform

$${\mathbb{E}}\left[e^{-\lambda R_{t}}\right]=e^{t\lambda\log\lambda},\quad\forall\lambda>0,\;t\geq 0,$$

such that $\{R_{t},t\geq 0\}$ is independent of $Z$. The main probabilistic result of this paper is the following theorem describing the limiting behavior of fluctuations of the extremal process. We use the notation $\Rightarrow$ for the convergence in distribution.

One can immediately deduce from the above theorem, that, conditionally on $Z$, for any test function $\phi_{0}\in{\mathcal{C}}^{+}_{bc}$, $\mathbb{L}_{Z}(\phi_{0})$ is a spectrally positive $1$-stable random variable with the Laplace transform:

$\displaystyle{\mathbb{E}}\left[e^{-\lambda\mathbb{L}_{Z}(\phi_{0})}\big{|}Z\right]=e^{Z\mu(\phi_{0})\lambda\log\lambda+\lambda Z\mu(\phi_{0})\log\mu(\phi_{0})-\lambda{Z}(m_{1}\mu(\phi_{0})+\nu(\phi_{0}))},\;\forall\lambda>0,$

(1.47)

We note that, in the study of BBM, both the $1$-stable process behavior and a small deterministic correction have appeared in a related but different context for the convergence (1.12) of $Z_{t}$ to $Z$ as $t\to+\infty$ in [21]. There, the correction is of the form $\log t/\sqrt{t}$, and is close in spirit to $e_{n}\sim\log n/n$ due to the diffusive nature of the space-time scaling, though the exact translation of the corrections appearing here and in [21] is not quite clear. The $\log t/t$ correction to the front location has also been observed in [5] and proved in [13]. The $1$-stable like tails for $Z$ itself have been observed already in [4] for the BBM, and in [20] for the branching random walk.

The paper is organized is follows. Section 2 explains how the the probabilistic statements, Theorem 1.2 and Theorem 1.5, follow from the correspomnding asymptoptics for the Bramson shift $x_{\varepsilon}$ in Proposition 1.1 and Theorem 1.4, as well as the the duality identities (1.14) and (1.15). The rest of the paper is devoted to the proof of Proposition 1.1 and Theorem 1.4. Section 3 introduces the self-similar variables that are used throughout the proofs, and reformulates the required results as Propositions 3.2 and 3.3. We also explain in that section how the values of the specific constants that appear in the asymptotics for $x_{\varepsilon}$ come about. Section 4 contains the analysis of the linear Dirichlet problem that appears throughout the PDE approach to the Bramson shift [14, 22, 23]. More specifically, we describe an approximate solution to the adjoint linear problem that is one reason for the logarithmic correction in $e_{n}$ in (1.40). We should stress that this is not the only contribution to the $\log n/n$ term – the second one comes from the nonlinear term in the Fisher-KPP equation. Section 5 contains the proof of Proposition 3.2. Of course, this Proposition is just a weaker version of Proposition 3.3, in the same vein as Proposition 1.1 is an immediate consequence of Theorem 1.4. However, the proof of Proposition 3.2 is much shorter than that of Proposition 3.3, so we present it separately for the convenience of the reader. Section 6 contains the proof of Proposition 3.3, with some intermediate steps proved in Section 7. Finally, Appendix A contains an auxiliary lemma on the continuity of the Bramson shift.

We use the notation $C$, $C^{\prime}$, $C_{0}$, $C_{0}^{\prime}$, etc. for various constants that do not depend on $\varepsilon$, and can change from line to line.

Acknowledgment. We are deeply indebted to Lisa Hartung for illuminating discussions during the early stage of this work, and are extremely grateful to Éric Brunet and Julien Berestycki for generously sharing their deep understanding of various aspects of the BBM and the Bramson shift, without which this work would not be possible. The work of JMR was supported by from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) ERC Grant Agreement n. 321186 - ReaDi, and from the ANR NONLOCAL project (ANR-14-CE25-0013). The work or LM and LR was supported by a US-Israel BSF grant, LR was partially supported by NSF grants DMS-1613603 and DMS-1910023, and ONR grant N00014-17-1-2145.

We first briefly explain the duality identities (1.14) and (1.15), and, in particular, the fact that no extra shift of the wave is needed in these relations, as soon as the wave normalization (1.6) is fixed. To see this, note that in the formal limit $\psi(x)\to\infty\cdot\mathbbm{1}(x\leq y)$, we have $\hat{\psi}(x)=\mathbbm{1}(x\leq y)$, and (1.15) reduces it to

$$\mathbb{P}[\chi_{1}\leq y]=1-U(\hat{s}[\chi_{0}]+y),~{}~{}\chi_{0}(x)=\mathbbm{1}(x\leq 0).$$

(2.1)

This is simply the definition of the Bramson shift, combined with the probabilistic interpretation (1.2) of the solution to (1.1) with the initial condition $u(0,x)=\mathbbm{1}(x\leq 0)$. Thus, no extra shift is needed neither in (2.1), nor in (1.15). The generalization of (2.1) to (1.15) is explained in Appendix C of [10]. Some normalization constants appear in the discussion there, but as we see, they are not needed once the wave is normalized by (1.6) rather than by

$$\int_{-\infty}^{\infty}x\tilde{U}^{\prime}(x)dx=0,$$

(2.2)

as in [10].

As for the derivative martingale identity (1.14), it is simply an immediate consequence of expressions (10) and (11) in [17], combined with the normalization (1.6) that implies that $C=1$ in both of these equations in [17].

We now prove Theorem 1.2. For an arbitrary $\phi_{0}\in\mathcal{C}^{+}_{bc}$, we set, as in (1.22),

$$\phi_{n}(x)=n^{-1}e^{-n}\phi_{0}(x-n),~{}~{}~{}\tilde{\phi}_{n}(x)=n^{-1}e^{-n}\phi_{0}(x),$$

(2.3)

and

$$\psi_{n}(x)=1-\exp\{-\phi_{n}(x)\},~{}~{}\tilde{\psi}_{n}(x)=1-\exp\{-\tilde{\phi}_{n}(x)\},$$

(2.4)

with

$$\psi_{0}(x)=1-\exp\{-\phi_{0}(x)\}.$$

(2.5)

Note that both $\psi_{n}$ and $\tilde{\psi}_{n}$ look like ”small step” initial conditions:

$$\psi_{n}(x)\approx\phi_{n}(x),~{}~{}\tilde{\psi}_{n}(x)\approx\tilde{\phi}_{n}(x)\hbox{ as $n\to+\infty$}.$$

We need the following result that, in fact, follows easily from [1], [3] and Appendix C in [10]. We provide the proof for the sake of completeness we provide the proof.

Proof. Fix an arbitrary $\phi_{0}\in\mathcal{C}^{+}_{c}$ and let $\{\mathcal{F}_{t}\}_{t\geq 0}$ be a filtration generated by $\{\mathcal{X}_{t}\}_{t\geq 0}$. Then, by the definition (1.10) of $\{\mathcal{X}_{t}\}_{t\geq 0}$, the Markov property and (1.3), we get (with some ambiguity of notation we set ${\mathbb{E}}={\mathbb{E}}_{0}$: the expectation for BBMs starting with one particle at $0$)

	$\displaystyle{\mathbb{E}}\Big{[}e^{-\mathcal{X}_{t}(\phi_{0})}$	$\displaystyle|\mathcal{F}_{s}\Big{]}={\mathbb{E}}\Big{[}\prod_{j=1}^{N_{t}}e^{-\phi_{0}(2t-(3/2)\log t-x_{j}(t)}|\mathcal{F}_{s}\Big{]}$		(2.8)
		$\displaystyle=\prod_{j=1}^{N_{s}}{\mathbb{E}}_{x_{j}(s)}\Big{[}\prod_{i=1}^{N_{t-s}}\Big{(}1-(1-e^{-\phi_{0}(2(t-s)-({3}/{2})\log(t-s)-x_{i}(t-s)+2s+({3}/{2})\log((t-{s})/{t}))}\Big{)}\Big{]}$
		$\displaystyle=\prod_{j=1}^{N_{s}}(1-u(t-s,m(t-s)-x_{j}(s)+2s+\frac{3}{2}\log(1-\frac{s}{t}))),\quad{\mathbb{P}}-\text{a.s.}$

Here, $u(t,x)$ is the solution to (1.1) with the initial condition $u(0,x)=\psi_{0}(x)=1-\exp(-\phi_{0}(x))$. Next, we use (1.8) to get

	$\displaystyle\lim_{t\rightarrow\infty}$	$\displaystyle\prod_{j=1}^{N_{s}}(1-u(t-s,m(t-s)-x_{j}(s)+2s+\frac{3}{2}\log(1-\frac{s}{t})))=\prod_{j=1}^{N_{s}}(1-U(\hat{s}[\psi_{0}]-x_{j}(s)+2s))$		(2.9)
		$\displaystyle=\exp\Big{\{}-\sum_{j=1}^{N_{s}}-\log(1-U(\hat{s}[\psi_{0}]-x_{j}(s)+2s)\Big{\}}.$

Following the derivations in (2.4.9)-(2.4.12) in [6] (see also (23)-(25) in [17]) and using (1.6) which gives $C=1$ in the above references we obtain

$$\lim_{s\rightarrow\infty}\exp\Big{\{}-\sum_{j=1}^{N_{s}}-\log(1-U(\hat{s}[\psi_{0}]-x_{j}(s)+2s)\Big{\}}=\exp\Big{\{}-Ze^{-\hat{s}[\psi_{0}]}\Big{\}},\quad{\mathbb{P}}-\text{~{}}{\rm a.s.}$$

(2.10)

Fix an arbitrary bounded continuous function $h$. Putting together (LABEL:eq:28_05_1), (LABEL:eq:28_05_2), and (2.10), we get,

	$\displaystyle\lim_{s\rightarrow\infty}$	$\displaystyle\lim_{t\rightarrow\infty}{\mathbb{E}}\Big{[}h(Z_{s})e^{-\mathcal{X}_{t}(\phi_{0})}\Big{]}=\lim_{s\rightarrow\infty}\lim_{t\rightarrow\infty}{\mathbb{E}}\Big{[}h(Z_{s}){\mathbb{E}}\Big{[}e^{-\mathcal{X}_{t}(\phi_{0})}|\mathcal{F}_{s}\Big{]}\Big{]}$		(2.11)
		$\displaystyle=\lim_{s\rightarrow\infty}{\mathbb{E}}\Big{[}h(Z_{s})e^{-\sum_{j=1}^{N_{s}}-\log(1-U(\hat{s}[\psi_{0}]-x_{j}(s)+2s)}\Big{]}={\mathbb{E}}\Big{[}h(Z)e^{-Ze^{-\hat{s}[\psi_{0}]}}\Big{]}.$

We used the bounded convergence theorem in the last equality.

Recall, that by Theorem 2.1 in [1], the pair $(\bar{}\mathcal{X}_{t},Z_{t})$ converges in distribution to $(\bar{}\mathcal{X},Z)$ as $t\rightarrow\infty$, where $\bar{}\mathcal{X}_{t}$ (resp. $\bar{}\mathcal{X}$) is just the measure $\mathcal{X}_{t}$ (resp. $\mathcal{X}$) shifted by $\log Z$. This together with a.s. convergence of $Z_{t}$ to $Z$ implies also convergence in distribution of $(\mathcal{X}_{t},Z_{t})$ to $(\mathcal{X},Z)$ as $t\rightarrow\infty$, and thus

$\displaystyle{\mathbb{E}}\Big{[}h(Z)e^{-\mathcal{X}(\phi_{0})}\Big{]}=\lim_{t\rightarrow\infty}{\mathbb{E}}\Big{[}h(Z_{t})e^{-\mathcal{X}_{t}(\phi_{0})}\Big{]}.$

From this we get

		$\displaystyle\left|{\mathbb{E}}\Big{[}h(Z)e^{-\mathcal{X}(\phi_{0})}\Big{]}-{\mathbb{E}}\Big{[}h(Z)e^{-Ze^{-\hat{s}[\psi_{0}]}}\Big{]}\right|=\left|\lim_{t\rightarrow\infty}{\mathbb{E}}\Big{[}h(Z_{t})e^{-\mathcal{X}_{t}(\phi_{0})}\Big{]}-{\mathbb{E}}\Big{[}h(Z)e^{-Ze^{-\hat{s}[\psi_{0}]}}\Big{]}\right|$
		$\displaystyle\leq\left|\lim_{t\rightarrow\infty}{\mathbb{E}}\Big{[}(h(Z_{t})-h(Z_{s}))e^{-\mathcal{X}_{t}(\phi_{0})}\Big{]}\right|+\left|\lim_{t\rightarrow\infty}{\mathbb{E}}\Big{[}h(Z_{s})e^{-\mathcal{X}_{t}(\phi_{0})}\Big{]}-{\mathbb{E}}\Big{[}h(Z)e^{-Ze^{-\hat{s}[\psi_{0}]}}\Big{]}\right|,\;\forall s>0.$

Therefore, we have

	$\displaystyle\left|{\mathbb{E}}\Big{[}h(Z)e^{-\mathcal{X}(\phi_{0})}\Big{]}-{\mathbb{E}}\Big{[}h(Z)e^{-Ze^{-\hat{s}[\psi_{0}]}}\Big{]}\right|$	$\displaystyle\leq\limsup_{s\rightarrow\infty}\lim_{t\rightarrow\infty}{\mathbb{E}}\Big{[}\left|h(Z_{t})-h(Z_{s})\right|\Big{]}$
		$\displaystyle+\left|\limsup_{s\rightarrow\infty}\lim_{t\rightarrow\infty}{\mathbb{E}}\Big{[}h(Z_{s})e^{-\mathcal{X}_{t}(\phi_{0})}\Big{]}-{\mathbb{E}}\Big{[}h(Z)e^{-Ze^{-\hat{s}[\psi_{0}]}}\Big{]}\right|=0,$

where convergence to zero of the first term follows from a.s. convergence of $Z_{t}$ to $Z$ and the bounded convergence theorem. The second term converges to zero by (2.11). Hence, we have

$${\mathbb{E}}\Big{[}h(Z)e^{-\mathcal{X}(\phi_{0})}\Big{]}={\mathbb{E}}\Big{[}h(Z)e^{-Ze^{-\hat{s}[\psi_{0}]}}\Big{]},$$

for any $\phi_{0}\in\mathcal{C}^{+}_{c}$ and any bounded continuous function $h$, which implies (2.6) for any $\phi_{0}\in\mathcal{C}^{+}_{c}$. The extension to $\phi_{0}\in\mathcal{C}^{+}_{bc}$ follows via approximation and a kind of continuity of $\hat{s}[\psi]$ in functions in $\mathcal{C}^{+}_{bc}$ and bounded by $1$, made precise in Lemma A.1 in Appendix A. $\Box$

We continue the proof of Theorem 1.2. Recall that we fixed an arbitrary $\phi_{0}\in\mathcal{C}^{+}_{bc}$. It follows from Proposition 1.1, with $\varepsilon_{n}=n^{-1}\exp(-n)$ that the Bramson shift appearing in the right side of (2.7) has the asymptotics

$$\hat{s}[\psi_{n}]=-n+\hat{s}[\tilde{\psi}_{n}]=-n+\log(ne^{n})-\log(n+\log n)-\log\bar{c}+o(1)\to-\log\mu(\phi_{0}),\hbox{ as $n\to+\infty$,}$$

(2.12)

with the measure $\mu$ defined in (1.28), and $\tilde{\psi}_{n}$ as in (2.4).

To get the weak convergence of the measures $Y_{n}(dx)$, it is sufficient to get the convergence of $Y_{n}(\phi_{0})$ for any $\phi_{0}\in\mathcal{C}^{+}_{bc}$. Thus, we will check convergence of the Laplace transforms of $Y_{n}(\phi_{0})$ (conditionally on $Z$). As a consequence of Lemma 2.1 and (2.12), we obtain

$$\lim_{n\rightarrow\infty}{\mathbb{E}}\left[e^{-Y_{n}(\phi_{0})}|Z\right]=\lim_{n\rightarrow\infty}e^{-Ze^{-\hat{s}[\psi_{n}]}}=e^{-Z\mu(\phi_{0})}.$$

(2.13)

Since $\phi_{0}\in\mathcal{C}_{bc}^{+}$ was arbitrary, this implies that, conditionally on $Z$,

$$Y_{n}\Rightarrow Z\mu,\quad\hbox{in ${\mathcal{M}}^{+}_{v}$},\;\mbox{as}\;n\rightarrow\infty.$$

(2.14)

Therefore, conditionally on $Z$, we have

$$Y_{n}(dx)\rightarrow Z\mu(dx)~{}~{}\hbox{in ${\mathcal{M}}^{+}_{v}$},\;\mbox{as}\;n\rightarrow\infty,\hbox{in probability,}$$

(2.15)

finishing the proof of Theorem 1.2. $\Box$