1. Introduction

We consider a supercritical branching Lévy process. At time 0, we start with a single particle which moves according to a Lévy process $\{(\xi_t)_{t\ge 0},{\rm P}\}$ with Lévy exponent $\psi(\theta)=\log {\rm E}\big({\textrm{e}}^{{\textrm{i}} \theta \xi_1}\big)$ . The lifetime of each particle is exponentially distributed with parameter $\beta$ , then it splits into k new particles with probability $p_k$ , $k\ge0$ . Once born, each particle will evolve independently, from its parent’s place of death, according to the same law as its parent, i.e. move according to the same Lévy process, and branch with the same branching rate and offspring distribution. We use $\mathbb{P}$ to denote the law of the branching Lévy process. Expectations with respect to $\mathbb{P}$ and P will be denoted by $\mathbb{E}$ and E respectively.

In this paper, we use ‘:=’ to denote a definition. For $a, b\in\mathbb{R}$ , $a \wedge b \,:\!=\, \min\{a, b\}$ . We will label each particle using the classical Ulam–Harris system. We write $\mathbb{T}$ for the set of all the particles in the tree, and o for the root of the tree. We also use the following notation:

• For any $u\in \mathbb{T}$ , $I_u^0$ denotes set of all the ancestors of u, $I_u\,:\!=\,I_u^0\cup\{u\}$ , and $n^u$ is the number of particles in $I_u\setminus \{o\}$ .

• For any $u\in \mathbb{T}$ , $\tau_u$ is the life length of u. Then $\{\tau_u,u\in\mathbb{T}\}$ are independent and identically distributed (i.i.d.), and exponentially distributed with parameter $\beta$ . Let $b_u$ and $\sigma_u$ be the birth and death times of u respectively. It is clear that $b_u=\sum_{v\in I_u^0}\tau_v$ and $\sigma_u=b_u+\tau_u$ . For any $t\ge 0$ , let $\mathcal{F}_t^{\mathbb{T}}\, :\!=\, \sigma\{b_u\wedge t,\sigma_u\wedge t\colon u\in\mathbb{T}\}$ .

• For any $t\ge 0$ , let $\mathcal{L}_t$ be the set of all particles alive at time t.

• Let $\{(X^u_s)_{s\ge 0}, u\in\mathbb{T}\}$ be i.i.d. with the same law as $\{(\xi_s)_{s\ge 0},{\rm P}\}$ and also independent of $\{\tau_u,u\in\mathbb{T}\}$ .

• For $u\in\mathcal{L}_t$ , let $\xi^u_t$ be the position of u at time t. Then, for $t\in [0, \sigma_o]$ , $\xi^o_t=X^o_t$ and, for any other $u\in \mathbb{T}$ ,

  (1.1) \begin{equation} \xi^u_t=\xi^{\pi(u)}_{\sigma_{\pi(u)}}+X^u_{t-b_u}=\sum_{v\in I_u^0}X^v_{\tau_v}+X^u_{t-b_u}, \qquad t\in [b_u, \sigma_u], \end{equation}
  where $\pi(u)$ denotes the parent of u.

• For $t\ge 0$ , $v\in \mathcal{L}_t$ , and $u\in I_v$ , we set $X_{u, t}\,:\!=\,\xi^u_{\sigma_u\wedge t}-\xi^u_{b_u\wedge t}$ . Note that $X_{v, t}=X^v_{t-b_v}$ and $X_{u,t}=X^u_{\tau_u}$ for all $u\in I^0_v$ .

For $t\geq 0$ , define $\mathbb{X}_t\,:\!=\,\sum_{u\in\mathcal{L}_t}\delta_{\xi^u_t}$ . The measure-valued process $(\mathbb{X}_t)_{t\geq 0}$ is called a branching Lévy process. When $\{(\xi_t)_{t\ge 0}, {\rm P}\}$ is a Brownian motion, $(\mathbb{X}_t)_{t\geq 0}$ is called a branching Brownian motion.

Denote by $Z_t$ the number of particles alive at time t. It is well known that $(Z_t)_{\geq 0}$ is a continuous-time branching process. In this paper we consider the supercritical case, i.e. $m\,:\!=\,\sum_k kp_k>1$ . Then $\mathbb{P} (\mathcal{S})>0$ , where $\mathcal{S}$ is the event of survival. The extinction probability $\mathbb{P} (\mathcal{S}^\textrm{c})$ is the smallest root in (0,1) of the equation $\sum_{k}p_k s^k =s$ ; see, for instance, [Reference Athreya and Ney6, Section III.4]. Define

(1.2) \begin{equation} \lambda\,:\!=\,\beta(m-1).\end{equation}

The process $\big({\textrm{e}}^{-\lambda t}Z_t\big)_{t\ge 0}$ is a non-negative martingale and hence

(1.3) \begin{equation} \lim_{t\to\infty}{\textrm{e}}^{-\lambda t}Z_t \,=\!:\,W \quad \text{exists almost surely (a.s.).}\end{equation}

For any two functions f and g on $[0,\infty)$ , $f\sim g$ as $s\to 0_+$ means that $\lim_{s\downarrow 0}({f(s)}/{g(s)})=1$ . Similarly, $f\sim g$ as $s\to \infty$ means that $\lim_{s\to\infty}({f(s)}/{g(s)})=1$ . Throughout this paper we assume the following two conditions hold.

The first condition is that the offspring distribution satisfies the Kesten–Stigum condition:

1. (H1) $\sum_{k\ge 1} (k\log k) p_k<\infty$ .

Condition (H1) ensures that W is non-degenerate with $\mathbb{P} (W>0)=\mathbb{P} (\mathcal{S})$ . For more details, see [Reference Athreya and Ney6, Section III.7].

The second condition is on the spatial motion:

1. (H2) There exist a complex constant $c_*$ with ${\textrm{Re}}(c_*)>0$ , $\alpha\in(0,2)$ , and a function $L(x)\,:\,\mathbb{R}_+\to \mathbb{R}_+$ slowly varying at $\infty$ such that $\psi(\theta)\sim -c_* \theta^{\alpha}L\big(\theta^{-1}\big)$ as $\theta\to 0_+$ .

Since ${\textrm{e}}^{\psi(\theta)}={\rm E}\big({\textrm{e}}^{{\textrm{i}} \theta \xi_1}\big)$ , we have ${\textrm{Re}}(\psi)\le 0$ and $\psi({-}\theta)=\overline{\psi(\theta)}$ . Thus, $\psi(\theta)\sim -\overline{c_*} |\theta|^{\alpha}L\big(|\theta|^{-1}\big)$ as $\theta\to 0_-$ . Under condition (H2), we can prove (see Remark 2.1) that ${\rm P} (|\xi_s|\ge x)\sim c s x^{-\alpha}L(x)$ as $x\to\infty$ , i.e. $|\xi_s|$ has regularly varying tails.

An important example satisfying (H2) is the strictly stable process.

Example 1.1. (Stable process.) Let $\xi$ be a strictly $\alpha$ -stable process, $\alpha\in (0,2)$ , on $\mathbb{R}$ with Lévy measure

$$n({\textrm{d}} y) = c_1x^{-(1+\alpha)}{\textbf{1}}_{(0,\infty)}(x)\,{\rm d}x + c_2|x|^{-(1+\alpha)}{\textbf{1}}_{({-}\infty,0)}(x)\,{\rm d}x,$$

where $c_1,c_2\ge 0$ , $c_1+c_2>0$ , and if $\alpha=1$ , $c_1=c_2=c$ . For $\alpha\in(1,2)$ , by [Reference Sato36, Lemma 14.11, (14.19)] and the fact that $\Gamma({-}\alpha)=-\alpha\Gamma(1-\alpha)$ , we obtain that, for $\theta>0$ ,

\begin{equation*} \int_0^\infty\big({\textrm{e}}^{{\textrm{i}} \theta y}-1-{\textrm{i}} \theta y\big)\,n({\rm d}y) = -c_1\alpha\Gamma(1-\alpha){\textrm{e}}^{-{\textrm{i}} \pi\alpha/2}\theta^\alpha, \end{equation*}

and, taking the conjugate on both sides of [Reference Sato36, Lemma 14.11 (14.19)], we get that

\begin{equation*} \int_{-\infty}^0\big({\textrm{e}}^{{\textrm{i}} \theta y}-1-{\textrm{i}} \theta y\big)\,n({\rm d}y) = -c_2\alpha\Gamma(1-\alpha){\textrm{e}}^{{\textrm{i}} \pi\alpha /2}\theta^\alpha. \end{equation*}

Thus, the Lévy exponent of $\xi$ is given, for $\theta>0$ , by

\begin{equation*} \psi(\theta) = \int \big({\textrm{e}}^{{\textrm{i}} \theta y}-1-{\textrm{i}} \theta y\big)\, n({\rm d} y) = -\alpha\Gamma(1-\alpha)\big(c_1{\textrm{e}}^{-{\textrm{i}} \pi\alpha/2}+c_2{\textrm{e}}^{{\textrm{i}} \pi\alpha/2}\big) \theta^\alpha. \end{equation*}

Similarly, by [Reference Sato36, Lemma 14.11 (14.18), (114.20)], we have, for $\theta>0$ ,

(1.4) \begin{align} \psi(\theta) & = \left\{ \begin{array}{l@{\quad}l} \displaystyle\int\big({\textrm{e}}^{{\textrm{i}} \theta y}-1\big)\,n({\rm d}y), & \alpha\in (0,1), \\ \displaystyle\int({\textrm{e}}^{{\textrm{i}} \theta y}-1-{\textrm{i}} \theta y{\textbf{1}}_{|y|\le 1})\,n({\rm d}y)+{\textrm{i}} a\theta, & \alpha=1 \end{array}\right. \nonumber \\ & = \left\{ \begin{array}{ll} -\alpha\Gamma(1-\alpha)\big(c_1{\textrm{e}}^{-{\textrm{i}} \pi\alpha/2}+c_2{\textrm{e}}^{{\textrm{i}} \pi\alpha/2}\big)\theta^\alpha, &\quad \alpha\in(0,1), \\ -c\pi\theta+{\textrm{i}} a\theta, & \quad \alpha=1, \end{array}\right. \end{align}

where $a\in\mathbb{R}$ is a constant. It is clear that $\psi$ satisfies (H2). For more details on stable processes, we refer the reader to [Reference Sato36, Section 14].

In Section 4 we give more examples satisfying condition (H2). Note that the non-symmetric 1-stable process does not satisfy (H2). However, in Example 4.1 we show that our main result still holds for the non-symmetric 1-stable process.

The maximal position $M_t\,:\!=\,\sup_{u\in\mathcal{L}_t}\xi^u_t$ of branching Brownian motions has been studied intensively. Assume that $\beta=1$ , $p_0=0$ , and $m=2$ . The seminal paper [Reference Kolmogorov29] proved that $M_t/t\to \sqrt{2}$ in probability as $t\to\infty$ . [Reference Bramson15] (see also [Reference Bramson16]) proved that, under some moment conditions, $\mathbb{P} (M_t-m(t)\le x)\to 1-w(x)$ as $t\to\infty$ for all $x\in \mathbb{R}$ , where $m(t)=\sqrt{2}t-{3}/{2\sqrt{2}}\log t$ and w(x) is a traveling wave solution. For more works on $M_t$ , see [Reference Chauvin and Rouault19, Reference Chauvin and Rouault20, Reference Lalley and Sellke30, Reference Roberts35]. For inhomogeneous branching Brownian motions, many papers have discussed the growth rate of the maximal position; see [Reference Bocharov12–Reference Bocharov and Harris14] for the case with catalytic branching at the origin, and [Reference Lalley and Sellke31, Reference Lalley and Sellke32, Reference Nishimori and Shiozawa34, Reference Shiozawa37] for the case with some general branching mechanisms.

Recently, the full statistics of the extremal configuration of branching Brownian motion have been studied. [Reference Arguin, Bovier and Kistler3, Reference Arguin, Bovier and Kistler4] studied the limit property of the extremal process of branching Brownian motion, proving that the random measure defined by $\mathcal{E}_t\,:\!=\,\sum_{u\in\mathcal{L}_t}\delta_{\xi^u_t-m(t)}$ converges weakly, and that the limiting process is a (randomly shifted) Poisson cluster process. Almost at the same time, [Reference Aïdékon, Berestycki, Brunet and Shi2] proved similar results using a totally different method.

For branching random walks, several authors have studied similar problems under an exponential moment assumption on the displacements of the offspring from the parent [Reference Aïdékon1, Reference Carmona and Hu18, Reference Hu and Shi27, Reference Madaule33]. When the displacements of the offspring from the parents are i.i.d. and have regularly varying tails, [Reference Durrett23] studied the limit property of its maximum displacement $M_n$ . More precisely, [Reference Durrett23] proved that $a_n^{-1}M_n$ converges weakly, where $a_n=m^{n/\alpha}L_0(m^n)$ and $L_0$ is slowly varying at $\infty$ . Recently, the extremal processes of the branching random walks with regularly varying steps were studied in [Reference Bhattacharya, Hazra and Roy8, Reference Bhattacharya, Hazra and Roy9], where it was proved that the point random measure $\sum_{|v|=n}\delta_{a_n^{-1}S_v}$ , where $S_v$ is the position of v, converges weakly to a Cox cluster process, which is quite different from the case with exponential moments. See also [Reference Bhattacharya, Maulik, Palmowski and Roy10, Reference Gantert25] for related works on branching random walks with heavy-tailed displacements.

[Reference Shiozawa, Chen, Takeda and Uemura38] studied branching symmetric stable processes with branching rate $\mu$ being a measure on $\mathbb{R}$ in a Kato class with compact support (i.e. the support of $\mu$ is compact) and the offspring distribution $\{p_n(x), n\geq 0\}$ being spatially dependent. Under some conditions on $\mu$ and $\{p_n(x), n\geq 0\}$ , [Reference Shiozawa, Chen, Takeda and Uemura38] proved that the growth rate of the maximal displacement is exponential with rate given by the principal eigenvalue of the mean semigroup of the branching symmetric stable process. In this paper, we study the extremes of branching Lévy processes with constant branching rate $\beta$ (that is, $\mu({\textrm{d}} x)=\beta\, {\textrm{d}} x$ ) and spatial motion having regularly varying tails (see condition (H2)). Since $\beta\,{\textrm{d}} x$ is not compactly supported, we cannot get the growth rate of the maximal displacement from [Reference Shiozawa, Chen, Takeda and Uemura38]. As a corollary of our extreme limit result we get the growth rate of the maximal displacement, see Corollary 1.2.

The key idea of the proof in this paper is the ‘one large jump principle’ inspired by [Reference Bhattacharya, Hazra and Roy8, Reference Bhattacharya, Hazra and Roy9, Reference Durrett23]. Along the discrete times $n\delta$ , the branching Lévy process $\{\mathbb{X}_{n\delta},n\ge 1\}$ is a branching random walk and the displacements from parents has the same law as $\mathbb{X}_\delta$ . It is natural to think that we may get the results of this paper from the results for branching random walks by letting the time grid become finer and finer, and appropriately controlling the behavior between the time gaps. However, we cannot apply the results for branching random walks in [Reference Bhattacharya, Hazra and Roy8, Reference Bhattacharya, Hazra and Roy9, Reference Madaule33] to $\{\mathbb{X}_{n\delta},n\ge 1\}$ . First, under condition (H2), the exponential moment assumption in [Reference Madaule33] is not satisfied. Second, [Reference Bhattacharya, Hazra and Roy8] assumes that the displacements are i.i.d., while the atoms of the random measure $\mathbb{X}_\delta$ , being particles alive at time $\delta$ in our branching Lévy process, are not independent. Last, although the displacements of offspring coming from the same parent are allowed to be dependent in [Reference Bhattacharya, Hazra and Roy9], [Reference Bhattacharya, Hazra and Roy9, Assumption 2.5], where the displacements from parents are given by a special form [Reference Bhattacharya, Hazra and Roy9, (2.9) and (2.10)]), seems to be very difficult to check for $\mathbb{X}_\delta$ .

Branching Lévy processes are closely related to the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation when the classical Laplacian $\Delta$ is replaced by the infinitesimal generator of the corresponding Lévy process. For any $g\in C_\textrm{b}^+(\mathbb{R})$ , define $u_g(t,x)=\mathbb{E}\big(\exp\big\{{-}\sum_{v\in\mathcal{L}_t}g\big(\xi^v_t+x\big)\big\}\big)$ . By the Markov and branching properties, we have

(1.5) \begin{equation} u_g(t,x) = {\rm E}\big({\textrm{e}}^{-g(\xi_t+x)}\big) + {\rm E}\int_0^t \varphi\big(u_g\big(t-s,\xi_s+x\big)\big)\,\textrm{d}s,\end{equation}

where $\varphi(s)=\beta\big(\sum_k s^kp_k-s\big)$ . Then $1-u_g$ is a mild solution to

(1.6) \begin{equation} \partial_t u-\mathcal{A} u=-\varphi(1-u),\end{equation}

with initial data $u(0,x)=1-{\textrm{e}}^{-g(x)}$ , where $\mathcal{A}$ is the infinitesimal generator of $\xi$ . [Reference Cabré and Roquejoffre17] proved that, under the assumption that the density of $\xi$ is comparable to that of a symmetric $\alpha$ -stable process, the frontal position of $1-u$ is exponential in time. Using our main result, we give another proof of [Reference Cabré and Roquejoffre17, Theorem 1.5] and also partially generalize it; see Remark 5.1.

1.1. Main results

Put ${\mathbb{R}}_0=({-}\infty,\infty)\setminus \{0\}$ , and $\overline{\mathbb{R}}_0=[{-}\infty,\infty]\setminus\{0\}$ with the topology generated by the set { $(a,b)$ , $({-}b,-a), (a,\infty], [{-}\infty,-a) \,:\, 0<a<b\le \infty$ }. Note that, for any $a>0$ , $[a,\infty]$ and $[{-}\infty,-a]$ are compact subsets of $\overline{\mathbb{R}}_0$ . Denote by $\mathcal{B}_\textrm{b}^+\big(\overline{\mathbb{R}}_0\big)$ the set of all bounded non-negative Borel functions on $\overline{\mathbb{R}}_0$ . Let $C_\textrm{c}^+\big(\overline{\mathbb{R}}_0\big)$ be the set of all non-negative continuous functions on $\overline{\mathbb{R}}_0$ such that $g=0$ on $({-}\delta,0)\cup(0,\delta)$ for some $\delta>0$ . Denote by $\mathcal{M}\big(\overline{\mathbb{R}}_0\big) $ the set of all Radon measures endowed with the topology of vague convergence (denoted by $\overset{\textrm{v}}{\to}$ ), generated by the maps $\mu\to \int f \,{\textrm{d}}\mu$ for all $f\in C_\textrm{c}^+\big(\overline{\mathbb{R}}_0\big)$ . Then $\mathcal{M}\big(\overline{\mathbb{R}}_0\big)$ is a metrizable space, see [Reference Kallenberg28, Theorem 4.2, p. 112]. For any $g\in\mathcal{B}_\textrm{b}^+\big(\overline{\mathbb{R}}_0\big)$ and $\mu\in\mathcal{M}\big(\overline{\mathbb{R}}_0\big)$ , we write $\mu(g)\,:\!=\,\int_{\overline{\mathbb{R}}_0} g(x)\,\mu({\rm d} x)$ . A sequence of random elements $\nu_n$ in $\mathcal{M}\big(\overline{\mathbb{R}}_0\big)$ converges weakly to $\nu$ , denoted as $\nu_n\overset{\textrm{d}}{\to}\nu$ , if and only if, for all $g\in C_\textrm{c}^+\big(\overline{\mathbb{R}}_0\big)$ , $\nu_n(g)$ converges weakly to $\nu(g)$ .

We claim that there exists a non-decreasing function $h_t$ with $h_t\uparrow\infty$ such that

(1.7) \begin{equation} \lim_{t\to\infty}{\textrm{e}}^{\lambda t}h_t^{-\alpha}L(h_t)=1,\end{equation}

where $\lambda$ is defined by (1.2). In fact, using [Reference Bingham, Goldie and Teugels11, Theorem 1.5.4], there exists a non-increasing function g such that $g(x)\sim x^{-\alpha}L(x)$ as $x\to\infty$ . Then $g(x)\to 0$ as $x\to\infty$ . Define $h_t\,:\!=\,\inf\{x>0\,:\, g(x)\le {\textrm{e}}^{-\lambda t}\}$ . It is clear that $h_t$ is non-decreasing and $h_t\uparrow\infty$ . By the definition of $h_t$ , for any $\varepsilon>0$ , $g(h_t/(1+\varepsilon)) \ge {\textrm{e}}^{-\lambda t} \ge g(h_t(1+\varepsilon))$ , which implies that

\begin{align*} (1+\varepsilon)^{-\alpha} & = (1+\varepsilon)^{-\alpha}\lim_{t\to\infty}\frac{L(h_t)}{L(h_t/(1+\varepsilon))} = \lim_{t\to\infty}\frac{g(h_t)}{g(h_t/(1+\varepsilon))} \\ & \le \liminf_{t\to\infty}{\textrm{e}}^{\lambda t}g(h_t) \le \limsup_{t\to\infty}{\textrm{e}}^{\lambda t}g(h_t) \\ & \le \lim_{t\to\infty}\frac{g(h_t)}{g(h_t(1+\varepsilon))} = (1+\varepsilon)^{\alpha} \lim_{t\to\infty}\frac{L(h_t)}{L(h_t(1+\varepsilon))} = (1+\varepsilon)^{\alpha}.\end{align*}

Since $\varepsilon$ is arbitrary, we get $\lim_{t\to\infty}{\textrm{e}}^{\lambda t}h_t^{-\alpha}L(h_t)=\lim_{t\to\infty}{\textrm{e}}^{\lambda t} g(h_t)=1$ . In particular, $h_t={\textrm{e}}^{\lambda t/\alpha}$ if $L=1$ . In Lemma 2.1 we prove that ${\textrm{e}}^{\lambda t}{\rm P}(h_t^{-1}\xi_s\in\cdot)\overset{\textrm{v}}{\to } sv_\alpha({\cdot})$ , where

(1.8) \begin{equation} v_\alpha({\textrm{d}} x) = q_1 x^{-1-\alpha}{\textbf{1}}_{(0,\infty)}(x)\,{\rm d}x + q_2|x|^{-1-\alpha}{\textbf{1}}_{({-}\infty,0)}(x)\,{\rm d}x,\end{equation}

with $q_1$ and $q_2$ being non-negative numbers, uniquely determined by $c_*=\alpha\Gamma(1-\alpha)(q_1{\textrm{e}}^{-{\textrm{i}} \pi\alpha/2} + q_2{\textrm{e}}^{{\textrm{i}} \pi\alpha/2})$ if $\alpha\neq 1$ and $q_1=q_2={\textrm{Re}}(c_*)/\pi$ if $\alpha=1$ .

Now we are ready to state our main result. Define a renormalized version of $\mathbb{X}_t$ by

(1.9) \begin{equation} \mathcal{N}_t\,:\!=\,\sum_{v\in\mathcal{L}_t}\delta_{h_t^{-1}\xi_t^v}.\end{equation}

In this paper we will investigate the limit of $\mathcal{N}_t$ as $t\to\infty$ .

Theorem 1.1. Under $\mathbb{P}$ , $\mathcal{N}_t$ converges weakly to a random measure $\mathcal{N}_\infty \in \mathcal{M}\big(\overline{\mathbb{R}}_0\big)$ defined on some extension $(\Omega,{\mathcal{G}},P)$ of the probability space on which the branching Lévy process is defined, with Laplace transform given by

\begin{equation*} E\big({\textrm{e}}^{-\mathcal{N}_\infty(g)}\big) = \mathbb{E}\bigg(\exp\bigg\{{-}W\int_0^\infty{\textrm{e}}^{-\lambda r} \int_{\mathbb{R}_{0}}\mathbb{E}\big(1-{\textrm{e}}^{-Z_{r}g(x)}\big)\,v_\alpha({\textrm{d}} x)\,{\rm d}r\bigg\}\bigg), \quad g\in C_\textrm{c}^+\big(\overline{\mathbb{R}}_0\big), \end{equation*}

where $\lambda$ is defined in (1.2) and W is the martingale limit defined in (1.3). Moreover, $\mathcal{N}_\infty=\sum_{j}T_j\delta_{e_j}$ , where, given W, $\sum_{j}\delta_{e_j}$ is a Poisson random measure with intensity $\vartheta Wv_\alpha({\textrm{d}} x)$ , $\{T_j,j\ge 1\}$ is a sequence of i.i.d. random variables with common law

(1.10) \begin{equation} P(T_j=k) = \vartheta^{-1}\int_0^\infty{\textrm{e}}^{-\lambda r}\mathbb{P}(Z_r=k)\,{\rm d}r, \qquad k\ge1, \end{equation}

where $v_\alpha({\textrm{d}} x)$ is given by (1.8), $Z_r$ is the number of particles alive at time r, $\vartheta=\int_0^\infty{\textrm{e}}^{-\lambda r}\mathbb{P}(Z_r>0)\,{\rm d}r$ , and $\sum_{j}\delta_{e_j}$ and $\{T_j,j\ge 1\}$ are independent.

Theorem 1.1 says that, given W, $\mathcal{N}_\infty$ is an integer-valued random measure with the locations of the atoms being a Poisson random measure with intensity $\vartheta Wv_\alpha({\textrm{d}} x)$ and with weights being i.i.d. with common distribution given by (1.10).

The proof of Theorem 1.1 consists of two steps. First, we use the idea of ‘one large jump’, which has been used in [Reference Bhattacharya, Hazra and Roy8, Reference Bhattacharya, Hazra and Roy9, Reference Durrett23] for branching random walks, to deduce that $\mathcal{N}_t$ has the same limit as the family of random measures defined by

$$\widetilde{\mathcal{N}}_t\,:\!=\,\sum_{v\in\mathcal{L}_t}\sum_{u\in I_v}\delta_{h_t^{-1}X_{u,t}}.$$

By ‘one large jump’ we mean that with large probability, for all $v \in \mathcal{L}_t$ , at most one of the ancestors of v has a large enough movement. Then we prove that with large probability, for all u born before $t-s$ , $\big|h_t^{-1}X_{u,t}\big|$ is small. Thus, the main contribution to $\widetilde{\mathcal{N}}_t$ is

$$\widetilde{\mathcal{N}}_{s,t}\,:\!=\,\sum_{v\in\mathcal{L}_t}\sum_{u\in I_v, b_u>t-s}\delta_{h_t^{-1}X_{u,t}}.$$

Remark 1.1. Given a function f, we use $D_f$ to denote its set of discontinuity points. Then, by Theorem 1.1, $\mathcal{N}_t(f)\overset{\textrm{d}}{\to}\mathcal{N}_\infty(f)$ for any bounded measurable function f on $\overline{\mathbb{R}}_0$ with compact support satisfying $\mathcal{N}_{\infty}(D_f)=0$ P-a.s. Furthermore, for any $k\ge1$ ,

$$(\mathcal{N}_t(B_1),\mathcal{N}_t(B_2),\ldots,\mathcal{N}_t(B_k)) \overset{\textrm{d}}{\to} (\mathcal{N}_\infty(B_1),\mathcal{N}_\infty(B_2),\ldots,\mathcal{N}_\infty(B_k)),$$

where $\{B_j\}$ are relatively compact subsets of $\overline{\mathbb{R}}_0$ satisfying $\mathcal{N}_\infty(\partial B_j)=0$ , $j=1,\ldots,k$ , P-a.s. See [Reference Kallenberg28, Theorem 4.4] for a proof.

Now we list the positions of all particles alive at time t in decreasing order, $M_{t,1}\ge M_{t,2}\ge \cdots \ge M_{t,Z_t},$ and for $n>Z_t$ define $M_{t,n}\,:\!=\,-\infty$ . In particular, $M_{t,1}=\max_{v\in\mathcal{L}_t}\xi^v_t$ is the rightmost position of the particles alive at time t. Note that $v_\alpha(0,\infty)=\infty$ if and only if $q_1>0$ . By the definition of $\mathcal{N}_\infty$ in Theorem 1.1, we have that if $q_1=0$ then $P(\mathcal{N}_\infty(0,\infty)=0)=P\big(\sum_j {\textbf{1}}_{e_j>0}=0\big)=1$ . If $q_1>0$ then

$$P(\mathcal{N}_\infty(0,\infty)=\infty\mid\mathcal{S}) =P\big(\textstyle\sum_j {\textbf{1}}_{e_j>0}=\infty\mid\mathcal{S}\big)=1,$$

and since, for any $x>0$ , $v_\alpha(x,\infty)<\infty$ , we have

$$P(\mathcal{N}_\infty(x,\infty)<\infty\mid\mathcal{S}) =P\bigg(\sum_j {\textbf{1}}_{e_j>x}<\infty\mid\mathcal{S}\bigg)=1.$$

Thus, on the set $\mathcal{S}$ , we can order the atoms of $\mathcal{N}_\infty$ on $(0,\infty)$ in decreasing order: $M_{(1)}\ge M_{(2)}\ge \cdots \ge M_{(k)}\ge \cdots\to 0$ . On the set $\mathcal{S}^\textrm{c}$ , $\mathcal{N}_{\infty}$ is null; then we define $M_{(k)}=-\infty$ for $k\geq 1$ .

Define $\mathbb{P}^*({\cdot}) \,:\!=\, \mathbb{P}(\cdot\mid\mathcal{S})$ $(P^*({\cdot})\,:\!=\,P(\cdot\mid\mathcal{S}))$ and let $\mathbb{E}^*$ $(E^*)$ be the corresponding expectation. As a consequence of Theorem 1.1, we have the following corollary.

Corollary 1.1. If $q_1>0$ then, for any $n\ge1$ ,

$$ \big(h_t^{-1}M_{t,1},h_t^{-1}M_{t,2},\dots,h_t^{-1}M_{t,n};\, \mathbb{P}^*\big) \overset{\textrm{d}}{\to} \big(M_{(1)},M_{(2)},\dots,M_{(n)};\, P^*\big). $$

Moreover, $M_{(k)}>0$ , $k\geq 1$ , $P^*$ -a.s.

In particular, for the rightmost position $R_t\,:\!=\,M_{t,1}=\max_{v\in\mathcal{L}_t}\xi^v_t$ , we have the following result.

Corollary 1.2. If $q_1>0$ then $\big(h^{-1}_t R_t;\, \mathbb{P}^*\big) \overset{\textrm{d}}{\to} \big(M_{(1)};\, P^*\big)$ , where the law of $\big(M_{(1)};\, P^*\big)$ is given by

$$ P^*\big(M_{(1)} \le x\big) = \left\{ \begin{array}{l@{\quad}l} \mathbb{E}^*\big({\textrm{e}}^{-\alpha^{-1}q_1\vartheta W x^{-\alpha}}\big), & x>0, \\ 0, & x\le 0. \end{array}\right. $$

Proof. Using Corollary 1.1, $\big(h^{-1}_tR_t;\, \mathbb{P}^*\big)\overset{\textrm{d}}{\to}\big(M_{(1)};\, P^*\big),$ and $M_{(1)}>0$ $P^*$ -a.s. For any $x>0$ , $P^*(M_{(1)}\le x) = P^*(\mathcal{N}_\infty(x,\infty)=0) = P^*\big(\sum_j {\textbf{1}}_{(x,\infty)}(e_j)=0\big) = \mathbb{E}^*\big({\textrm{e}}^{-\vartheta W v_\alpha(x,\infty)}\big) = \mathbb{E}^*\big({\textrm{e}}^{-\alpha^{-1}q_1\vartheta W x^{-\alpha}}\big)$ . The proof is now complete.

Remark 1.2. Similarly, we can order the particles alive at time t in an increasing order: $L_{t,1}\le L_{t,2}\le \cdots\le L_{t,Z_t}$ . When $q_2=0$ , $P(\mathcal{N}_\infty({-}\infty,0)=0)=P\big(\sum_j {\textbf{1}}_{e_j<0}=0\big)=1$ . When $q_2>0$ , on the set $\mathcal{S}$ , we can order the atoms of $\mathcal{N}_\infty$ on $({-}\infty,0)$ as $L_{(1)}\le L_{(2)}\le \cdots \le L_{(k)}\le \cdots\to 0$ . Note that $\{M_{(k)},k\ge1\}$ and $\{L_{(k)},k\ge 1\}$ cover all the atoms of $\mathcal{N}_\infty$ . Similar to Corollaries 1.1 and 1.2, we have the following weak convergence of $(L_{t,1}, L_{t,2},\ldots,L_{t,n})$ : if $q_2>0$ then, for any $n\ge1$ ,

$$ \big(h_t^{-1}L_{t,1},h_t^{-1}L_{t,2},\dots,h_t^{-1}L_{t,n};\, \mathbb{P} ^*\big) \overset{\textrm{d}}{\to} \big(L_{(1)},L_{(2)},\dots , L_{(n)};\, P^*\big);\, $$

and the distribution of $L_{(1)}$ under $P^*$ is as follows: for any $x<0$ , $P^*(L_{(1)}\le x) = P^*(\mathcal{N}_\infty({-}\infty,x]>0) = P^*\big(\sum_j{\textbf{1}}_{({-}\infty,x]}(e_j)>0\big) = 1 - \mathbb{E}^*\big({\textrm{e}}^{-\vartheta W v_\alpha({-}\infty,x]}\big) = 1 - \mathbb{E}^*\Big({\textrm{e}}^{-\alpha^{-1}q_2\vartheta W |x|^{-\alpha}}\Big)$ .

The rest of the paper is organized as follows. In Section 2 we introduce the one large jump principle and give the proof of Theorem 1.1 based on Proposition 2.1, which will be proved in Section 2.3. The proof of Corollary 1.1 is given in Section 3. In Section 4 we give more examples satisfying condition (H2) and conditions which are weaker than (H2), but sufficient for the main result of this paper. We discuss the front position of the Fisher–KPP equation (1.6) in Section 5.