2. Basic notation

The set of real numbers is denoted by $\mathbb{R}$ and is viewed as the usual Euclidean space. As a measurable space, $\mathbb{R}$ is endowed with the Borel $\sigma$ -algebra. For scalar functions g(x) of a real x, we write $\|g\|_1 = \int_x |g(x)|dx$ for the $L_1$ -norm; g(x) is called c-Lipschitz if it is Lipschitz with constant $c \ge 0$ . Let $\mathcal{C}_b$ be the set of continuous bounded functions on $\mathbb{R}$ which are constant outside a closed interval (with one constant value to the ‘left’ of the interval, and possibly another constant value to the ‘right’ of it.)

For functions g(x) of a real x, we denote by $g(x+)$ and $g(x-)$ the right and left limits at x; a function g(x) is RCLL if it is right-continuous and has left limits at each x.

A function g of x may be written as either g(x) or $g_x$ . The notation $d_x g(x,t)$ for a multivariate function g(x,t), where $x\in \mathbb{R}$ , means the differential in x.

Denote by $\mathcal{M}$ the set of scalar RCLL non-decreasing functions $f=(f(x), x\in \mathbb{R})$ that are (proper) probability distribution functions, i.e., such that $f(x) \in [0,1]$ , $\lim_{x\downarrow -\infty} f(x) = 0$ , and $\lim_{x\uparrow \infty} f(x) = 1$ . For elements $f \in \mathcal{M}$ we use the terms distribution function and distribution interchangeably. The space $\mathcal{M}$ is endowed with the Lévy–Prokhorov metric (cf. [Reference Ethier and Kurtz8]) and the corresponding topology of weak convergence (which is equivalent to the convergence at every point of continuity of the limit); the weak convergence in $\mathcal{M}$ is denoted $\stackrel{w}{\rightarrow}$ . Subspaces of $\mathcal{M}$ that we define later inherit its metric. The inverse ( $\nu$ th quantile) of $f \in \mathcal{M}$ is $f_{\nu}^{-1} \doteq \inf\{y\,|\,f_y \ge \nu\}$ , $\nu\in [0,1]$ ; $f_1^{-1} =\infty$ when $f_y < 1$ for all y; we will also use the notation $q_\nu(f) \doteq f_{\nu}^{-1} $ . For a given $\nu\in (0,1)$ , set $\mathring{\mathcal{M}} = \{f \in \mathcal{M} \,|\, f_{\nu}^{-1} =0\}$ ; the parameter $\nu$ used in the definition of $\mathring{\mathcal{M}}$ will be clear from the context.

Unless explicitly specified otherwise, we use the following conventions regarding random elements and random processes. A measurable space is assumed to be equipped with the Borel $\sigma$ -algebra induced by the metric, which is clear from the context. A random process $Y(t), \,t\ge 0,$ always takes values in a complete separable metric space (clear from the context) and has RCLL sample paths. For a random process $Y(t), \,t\ge 0,$ we denote by $Y(\infty)$ the random value of Y(t) in a stationary regime (which will be clear from the context). The symbol $\Rightarrow$ signifies convergence of random elements in distribution; $\stackrel{\mathbb{P}}{\longrightarrow}$ means convergence in probability. For a condition/event A, ${\mathbf{I}}\{A\}=1$ if A holds, and ${\mathbf{I}}\{A\}=0$ otherwise.

The space $D([0,\infty), \mathbb{R})$ (resp. $D([0,\infty), \mathcal{M})$ ) is the Skorokhod space of RCLL functions on $[0,\infty)$ taking values in $\mathbb{R}$ [resp. $\mathcal{M}$ ], with the corresponding Skorokhod ( $J_1$ ) metric and topology (cf. [Reference Ethier and Kurtz8]); the symbol $\stackrel{J_1}{\rightarrow}$ denotes convergence in these spaces.

For a distribution $f \in \mathcal{M}$ and scalar function $h(x), x \in \mathbb{R}$ , we define $f h \doteq \int_{\mathbb{R}} h(x) df_x$ . In some cases it is important whether or not an integral includes the endpoints of the integration interval; $\int_{a+}^b h(x) df_x$ means that the integral excludes the left endpoint a, and $\int_{a}^{b+} h(x) df_x$ means that the integral includes the right endpoint b.

For scalar functions $h(x), x \in \mathcal{X}$ , with some domain $\mathcal{X}$ , $\|h\| = \sup_{x\in \mathcal{X}} |h(x)|$ is the sup-norm. When $G_k, G$ are operators mapping the space of such functions into itself, $\lim G_k h = Gh$ and $G_k h \to Gh$ mean uniform convergence: $\|G_k h - Gh\| \to 0$ .

Suppose we have a Markov process with state space $\mathcal{X}$ and transition function $P^t(x,H),$ $t\ge 0$ . Then $P^t$ , as an operator, is $P^t h(x) \doteq \int_y P^t(x,dy) h(y)$ , where h is a scalar function with domain $\mathcal{X}$ ; $I=P^0$ is the identity operator. The process (infinitesimal) generator B is

\begin{align*}Bh \doteq \lim_{t\downarrow 0} \frac{1}{t} [P^t - I] h.\end{align*}

The function h is within the domain of the generator B if Bh is well-defined.

We denote by $\bar J(x) = 1- J(x)$ the complementary cumulative distribution function of the jump size distribution. The Heaviside step function (or Dirac distribution concentrated at the point 0) is denoted by $\chi=(\chi_x = {\mathbf{I}}\{x \ge 0\}) \in \mathcal{M}$ .

3. System process definition

We already noted that, without loss of generality, we may assume $\mathbb{E} Z= 1$ ; otherwise, this condition can be achieved by rescaling space. Note also that, without loss of generality, we can and do assume $\mu=1$ ; otherwise, we can achieve this condition by rescaling time.

Let $f^n(t) =(f_x^n(t), \,x\in \mathbb{R})$ be the (random) empirical distribution of the particle locations at time t; that is, $f_x^n(t)$ is the fraction of particles located in $({-}\infty,x]$ at time t. Clearly, $f^n({\cdot})$ is a Markov process with the state space $\mathcal{M}^{(n)} \subset \mathcal{M}$ . Let us also consider a centered version of the process. Fix a number $\nu \in (0,1)$ , and let $\mathring{g}_x = g_{x+q_{\nu}(g)}, \,x \in \mathbb{R}.$ That is, $\mathring{g}$ is a version of g that is centered at its $\nu$ th quantile. Then $\mathring{f} {}^n(t)$ —the centered version of $f^n(t)$ —is also a Markov process, with state space $\mathring{\mathcal{M}} {}^{(n)} = \mathcal{M}^{(n)} \cap \mathring{\mathcal{M}}$ . This process is regenerative, with the regenerations occurring at time points when all particles ‘collapse’ to one location. A regeneration cycle has finite mean; indeed, for any time interval of a fixed length $\varepsilon>0$ , with probability at least some $\delta=\delta(\varepsilon,n)>0$ , at the end of the interval all particles will be at a single location. Therefore the process is positive recurrent, with unique stationary distribution. Denote by $\mathring{f} {}^n(\infty)$ a random element with distribution equal to the stationary distribution of $\mathring{f} {}^n({\cdot})$ .

Consider a version $f^n({\cdot})$ of the system process, such that the centered process $\mathring{f} {}^n({\cdot})$ is stationary. Specifically, consider the process $f^n({\cdot})$ with $f^n(0)$ equal in distribution to $\mathring{f} {}^n(\infty)$ . Then, for each n, the steady-state average speed $v_n$ of the particle system is finite and given by

\begin{align*}v_n = \frac{1}{T} \mathbb{E} [q_\beta(f^n(t)) - q_\beta(f^n(0))],\end{align*}

where $\beta\in (0,1)$ and $t>0$ can be chosen arbitrarily. Also, $v_n$ is the steady-state average speed of the center of mass, i.e. of the mean of the distribution $f^n(t)$ :

\begin{align*}v_n = \lambda + \mathbb{E} \int_{-\infty}^{\infty} d_y \mathring{f} {}_y^n(\infty) \int_y^\infty d_\xi \mathring{f} {}_{\xi}^n(\infty) (\xi-y).\end{align*}

Given the regenerative structure of the (centered) process, the steady-state speed $v_n$ can also be defined in terms of probability-one convergence, regardless of the initial state. In particular, for any initial state of the system,

\begin{align*}v_n = \lim_{t\to\infty} \frac{1}{t} [D^n(t) - D^n(0)], \,\,\text{with probability 1},\end{align*}

where $D^n(t)$ is the location of the leading (rightmost) particle.

We remark that, trivially, $\lambda=0$ implies $v_n=0$ , because then, with probability one, all particles will end up at a single location and will stop moving after that. Therefore, $v_n > 0$ if and only if $\lambda>0$ , and we have the trivial lower bound $v_n \ge \lambda$ for any n.

Recall that we assumed $\mu=1$ . If we do need expressions for a case when $\mu\ne 1$ , they are obtained from the corresponding expressions for the $\mu=1$ case. For example, if $v_n=v_n(\lambda,\mu)$ is the steady-state speed as a function of $\lambda$ and $\mu>0$ , then $v_n(\lambda,\mu) = \mu v_n(\lambda/\mu,1)$ .

4. Monotonicity properties

The order relation (stochastic dominance) $g^{(1)} \preceq g^{(2)}$ between two distributions $g^{(1)}, g^{(2)} \in \mathcal{M}$ means $g^{(1)}_x \ge g^{(2)}_x, \,\forall x$ .

The process $f^n({\cdot})$ possesses some simple monotonicity properties. If we have two versions of this process $f^n({\cdot})$ , labeled $f^{n,(1)}({\cdot})$ and $f^{n,(2)}({\cdot})$ , with fixed initial conditions $f^{n,(1)}(0) \preceq f^{n,(2)}(0)$ , then these two processes can be coupled (constructed on a common probability space) so that $f^{n,(1)}(t) \preceq f^{n,(2)}(t)$ at all times $t\ge 0$ . Indeed, it suffices to set a one-to-one correspondence between particles in the two systems, such that each particle in the second system in initially ahead of the corresponding particle in the first system, and consider a probability space such that each pair of corresponding particles has the same realizations of time instants and sizes of the independent jumps, as well as the same realizations of the time instants and ‘targets’ of the synchronization jumps. Then, clearly, each particle in the second system will remain ahead of its counterpart in the first system at all times.

This property easily generalizes to the case when the second (‘larger’) process has a larger number of particles, i.e. when we compare $f^{n,(1)}({\cdot})$ and $f^{k,(2)}({\cdot})$ , where $n \le k$ . In this case, we will use the order relation $f^{n,(1)}(t) \preceq_l f^{k,(2)}(t)$ , which denotes the situation when $f^{n,(1)}(t) \preceq f^{n(k),(2)}(t)$ , where $f^{n(k),(2)}(t)$ is the empirical distribution of the leading n particles of the state $f^{k,(2)}({\cdot})$ . Again, if $f^{n,(1)}(0) \preceq_l f^{k,(2)}(0)$ , then the processes can be coupled so that $f^{n,(1)}(t) \preceq_l f^{k,(2)}(t)$ prevails at all times. This extension has been used in the literature in different contexts (cf. the proof of Theorem 2.3(4) in [Reference Groisman, Jonckheere and Martínez13], and references therein), but is also easy to see directly. Indeed, the synchronization jumps can equivalently be defined as follows: a particle gets a ‘synchronization urge’ as a Poisson process of rate $\mu$ (rescaled to 1), and it relocates another particle, chosen uniformly at random, to its own location if the chosen particle happens to be behind it. Given this interpretation, the coupling construction is straightforward.

Since the process is such that (a) for any initial state, $v_n = \lim_{t\to\infty} D^n(t)/t$ almost surely, where $D^n(t)$ is the location of the leading particle at time t, and (b) monotonicity with respect to $\preceq_l$ holds, we see that the steady-state speed $v_n$ is non-decreasing in n:

(4.1) \begin{equation}v_n \le v_{n+1}, \,\,\forall n \ge 1.\end{equation}

(Again, this property was used in the proof of Theorem 2.3(4) in [Reference Groisman, Jonckheere and Martínez13] for a different system, but it obviously holds for any system satisfying (a) and (b).)

The monotonicity with respect to $\preceq$ (and, more generally, $\preceq_l$ ) easily generalizes further in several directions. For example, it still holds if the ‘larger’ process $f^{k,(2)}({\cdot})$ has larger parameter $\lambda$ , or if it is ‘helped’ by a left boundary (which can be static or moving in any way), or if the ‘smaller’ process $f^{n,(1)}({\cdot})$ is ‘impeded’ by a right boundary (which can be static or moving in any way), etc. We will not give formal statements for all these generalizations, and we will refer to the monotonicity properties described in this section simply as monotonicity.

5. Mean-field limits

In this section we state our results on the convergence to and characterization of MFLs. The results are for the original system, the system with fixed right boundary (which is primarily a tool for the analysis of the original system), and the two systems with moving right and left boundaries. We will use the same notation $f^n({\cdot})$ and $f({\cdot})$ for the process and an MFL, respectively, for each of the four systems. It should be clear that they do depend on the system, and when we use them later in the paper it will be clear from the context which system they refer to.

5.1. Original system

The following definition of a mean-field model describes what it is natural to expect an MFL to be. Theorem 1 then shows that MFLs this definition.

Definition 1. A function $f_x(t), \,x\in \mathbb{R}, \,t\in \mathbb{R}_+,$ will be called a mean-field model (MFM) if it satisfies the following conditions:

1. (a) For any t, $f(t) = (f_x(t), x\in \mathbb{R}) \in \mathcal{M}$ .

2. (b) For any x, $f_x(t)$ is non-increasing and $(\lambda+1)$ -Lipschitz in t.

3. (c) For any x, for any t where the partial derivative $(\partial/\partial t) f_x(t)$ exists (which is almost all t with respect to the Lebesgue measure, by the Lipschitz property), the equation

   (5.1) \begin{equation}\frac{\partial}{\partial t} f_x(t) = -\lambda \int_{-\infty}^x d_y f_y(t) (1-J(x-y)) - f_x(t) (1-f_x(t)).\end{equation}
   holds.

Denote by $L^{(n)}$ the generator of the process $f^n({\cdot})$ . For any $h\in\mathcal{C}_b$ , the function $f^n h$ of an element $f^n \in \mathcal{M}^{(n)}$ is within the domain of $L^{(n)}$ (where we use the fact that each function in $\mathcal{C}_b$ is constant outside a closed interval), and

\begin{align*}L^{(n)} [f^n h] =\lambda \mathbb{E}_Z \int_{-\infty}^{\infty} d_y f_y^n (h(y+Z)-h(y))+ \int_{-\infty}^{\infty} d_y f_y^n \int_y^\infty d_\xi f_{\xi}^n (h(\xi)-h(y)),\end{align*}

where the expectation is over the distribution of the random jump size Z. We also formally define the ‘limit’ L of $L^{(n)}$ (for elements $f \in \mathcal{M}$ ) as

\begin{align*}L [f h] =\lambda \mathbb{E}_Z \int_{-\infty}^{\infty} d_y f_y (h(y+Z)-h(y))+ \int_{-\infty}^{\infty} d_y f_y \int_y^\infty d_\xi f_{\xi} (h(\xi)-h(y)).\end{align*}

Theorem 1. Suppose $f^n(0) \stackrel{w}{\rightarrow} f(0)$ , where $\{f^n(0)\}$ is a deterministic sequence of elements $f^n(0) \in \mathcal{M}^{(n)}$ , and $f(0) \in \mathcal{M}$ . Then $f^n({\cdot}) \Rightarrow f({\cdot})$ in $D([0,\infty), \mathcal{M})$ , where $f({\cdot}) \in D([0,\infty), \mathcal{M})$ is deterministic and uniquely determined by f(0). Moreover, $f({\cdot})$ is a continuous element of $D([0,\infty), \mathcal{M})$ and satisfies

(5.2) \begin{equation}f(t) h - f(0) h - \int_0^t L f(s) h ds =0, \,\,\forall h\in \mathcal{C}_b, \,\forall t\ge 0.\end{equation}

The trajectory $f({\cdot})$ is called the mean-field limit (MFL) with initial state f(0). Furthermore, the MFL $f({\cdot})$ is the unique MFM (see Definition 1) with initial state f(0).

The proof of Theorem 1 is in Section 13.

The MFL $f({\cdot})$ with initial state $f(0)=\chi$ (i.e., the Dirac distribution concentrated at a single point 0) will be called the benchmark MFL (BMFL).

5.2. System with fixed right boundary

As an intermediate step towards the proof of Theorem 1, we will prove a special case of it, Theorem 2 below, which applies to the system with fixed right boundary, defined as follows.

Note that, for any fixed B, the evolution of $f_x^n({\cdot}), x \in ({-}\infty, B)$ in our original system is independent of the actual locations and/or evolution of particles in $[B, \infty)$ . Therefore, if we are only interested in the evolution of $f_x^n({\cdot}), x \in ({-}\infty, B)$ , we may as well assume that any particle in $[B, \infty)$ is located exactly at B (that is, $f_B^n(t)=1$ at all times). In other words, we can assume that there is a fixed right boundary point B, and any particle which ‘tries to jump over B’ lands (absorbed) at exactly B. This is the modified system (with fixed right boundary) that we consider here. We see that the process $f^n({\cdot})$ for this system is a projection of the original process—it is such that $f^n(t) \in \mathcal{M}^B \doteq \{g \in \mathcal{M} \,|\, g_B=1\}$ for all t.

Denote by $L^{(n),B}$ the generator of the process $f^n({\cdot})$ for this system. For any $h\in\mathcal{C}_b$ , the function $f^n h$ of an element $f^n \in \mathcal{M}^{(n)} \cap \mathcal{M}^B$ is within the domain of $L^{(n),B}$ , and

\begin{align*}L^{(n),B} [f^n h] =\lambda \mathbb{E} \int_{-\infty}^{B} d_y f_y^n (h((y+Z) \vee B)-h(y))+ \int_{-\infty}^{B} d_y f_y^n \int_y^{B+} d_\xi f_{\xi}^n (h(\xi)-h(y)),\end{align*}

where the expectation is over the distribution of the random jump size Z. We also formally define the ‘limit’ $L^{B}$ of $L^{(n),B}$ (for elements $f \in \mathcal{M} \cap \mathcal{M}^B$ ) as

\begin{align*}L^{B} [f h] =\lambda \mathbb{E} \int_{-\infty}^{B} d_y f_y (h((y+Z) \vee B)-h(y))+ \int_{-\infty}^{B} d_y f_y \int_y^{B+} d_\xi f_{\xi} (h(\xi)-h(y)).\end{align*}

Observe the following. For a fixed $h\in\mathcal{C}_b$ , if B is large enough so that h(x) is constant for all $x \ge B$ , then $L^{(n),B} [f^n h]= L^{(n)} [f^n h]$ and $L^{B} [f^n h]= L [f^n h]$ , where $L^{(n)}$ and L are the operators we defined for the original system.

Theorem 2. Suppose $f^n(0) \stackrel{w}{\rightarrow} f(0)$ , where $\{f^n(0)\}$ is a deterministic sequence of elements $f^n(0) \in \mathcal{M}^{(n)} \cap \mathcal{M}^B$ , and $f(0) \in \mathcal{M}^B$ . Then $f^n({\cdot}) \Rightarrow f({\cdot})$ in $D([0,\infty), \mathcal{M}^B)$ , where $f({\cdot}) \in D([0,\infty), \mathcal{M}^B)$ is deterministic and uniquely determined by f(0). Moreover, $f({\cdot})$ is a continuous element of $D([0,\infty), \mathcal{M}^B)$ which satisfies

(5.3) \begin{equation}f(t) h - f(0) h - \int_0^t L^B f(s) h ds =0, \,\,\forall h\in \mathcal{C}_b, \,\forall t\ge 0.\end{equation}

This trajectory $f({\cdot})$ is called the mean-field limit (MFL) with initial state f(0). Furthermore, the MFL $f({\cdot})$ is the unique MFM with initial state f(0); the MFM here is defined as in Definition 1 except that (5.1) holds for $x < B$ , and $f_x(t) \equiv 1$ for $x\ge B$ and all t.

The proof of Theorem 2 is in Section 12.

5.3. System with moving right boundary

Consider now the system with the right boundary point B moving to the right at constant speed $v>0$ , i.e. $B=B_0+vt$ for some constant $B_0$ . Just as in the system with fixed right boundary, any particle which ‘tries to jump over the boundary’ lands exactly at its current location. However, particles landing at the right boundary are not ‘absorbed’ in it, as the boundary keeps moving right while the particles do not move until they jump again. If B is the boundary location at time t, then the state $f^n(t) \in \mathcal{M}^B$ . Formally speaking, the state of the Markov process has an additional component corresponding to the real number B; somewhat abusing notation, we will not include this component in the state descriptor.

It is important to note that the process with moving right boundary is not a projection of the original process.

Denote by $L^{(n), \{r\}}$ the generator of the process $f^n({\cdot})$ for this system. For any $h\in\mathcal{C}_b$ , the function $f^n h$ of an element $f^n$ (or rather pair $(f^n,B)$ ) satisfying $f^n \in \mathcal{M}^{(n)} \cap \mathcal{M}^B$ is within the domain of $L^{(n), \{r\}}$ , and

\begin{align*}L^{(n), \{r\}} [f^n h] =\lambda \mathbb{E} \int_{-\infty}^{B} d_y f_y^n (h((y+Z) \vee B)-h(y))+ \int_{-\infty}^{B} d_y f_y^n \int_y^{B+} d_\xi f_{\xi}^n (h(\xi)-h(y)),\end{align*}

where the expectation is over the distribution of the random jump size Z. We also formally define the ‘limit’ $L^{\{r\}}$ of $L^{(n), \{r\}}$ (for pairs (f,B) satisfying $f \in \mathcal{M}^B$ ) as

\begin{align*}L^{\{r\}} [f h] =\lambda \mathbb{E} \int_{-\infty}^{B} d_y f_y (h((y+Z) \vee B)-h(y))+ \int_{-\infty}^{B} d_y f_y \int_y^{B+} d_\xi f_{\xi} (h(\xi)-h(y)).\end{align*}

The operators $L^{(n), \{r\}}$ and $L^{\{r\}}$ are the same as the operators $L^{(n), B}$ and $L^B$ , respectively, for the system with fixed right boundary at B. Here, however, B is part of the process state, rather than a fixed constant.

Theorem 3. Let $v>0$ and $B_0$ be fixed, and recall that the boundary is given by $B=B_0 +vt$ . Suppose $f^n(0) \stackrel{w}{\rightarrow} f(0)$ , where $\{f^n(0)\}$ is a deterministic sequence of elements $f^n(0) \in \mathcal{M}^{(n)} \cap \mathcal{M}^{B_0}$ , and $f(0) \in \mathcal{M}^{B_0}$ . Then $f^n({\cdot}) \Rightarrow f({\cdot})$ in $D([0,\infty), \mathcal{M})$ , where $f({\cdot}) \in D([0,\infty), \mathcal{M})$ is deterministic and uniquely determined by f(0); $f^n(t) \in \mathcal{M}^{(n)} \cap \mathcal{M}^{B}$ and $f(t) \in \mathcal{M}^{B}$ for all t. Moreover, $f({\cdot})$ is a continuous element of $D([0,\infty), \mathcal{M})$ which satisfies

(5.4) \begin{equation}f(t) h - f(0) h - \int_0^t L^{\{r\}} f(s) h ds =0, \,\,\forall h\in \mathcal{C}_b, \,\forall t\ge 0.\end{equation}

This trajectory $f({\cdot})$ is called the mean-field limit (MFL) with initial state f(0). Furthermore, MFL $f({\cdot})$ is the unique MFM with initial state f(0); the MFM here is defined as in Definition 1 except that (5.1) holds for $x < B (=B_0 + vt)$ .

The proof of Theorem 3 is in Section 14.

5.4. System with moving left boundary

Consider yet another modification of our original system, namely the system with left boundary A moving right at constant speed $v>0$ , i.e. $A=A_0+vt$ for some constant $A_0$ . In this system there are no particles to the left of A—as the boundary moves right it ‘drags forward’ any particle that it encounters, and each such particle stays at the (moving) boundary until it jumps forward. The corresponding process is not a projection of our original process.

By the definition of the process, there are no particles located to the left of the moving boundary A. However, for the purposes of analysis it will be convenient to adopt an equivalent view of the process which allows particles to be to the left of A. Specifically, assume that particles make synchronization jumps as in the original system, but when a particle located at y makes an independent jump, it first instantly moves to the point $y \vee A$ and then jumps. Clearly, this process is such that the evolution in time of $(f^n_x(t), x \ge A)$ is exactly the same as it would be if the moving left boundary dragged the particles that it encountered with it. So this is how we will define the process $f^n({\cdot})$ for this system. Then $f^n(t) \in \mathcal{M}^{(n)}$ for all t, and the boundary location A is implicitly a part of the definition of the process.

Denote by $L^{(n), \{l\}}$ the generator of the process $f^n({\cdot})$ . For any $h\in\mathcal{C}_b$ , the function $f^n h$ of an element $f^n$ (or rather pair $(f^n,A)$ ) satisfying $f^n \in \mathcal{M}^{(n)}$ is within the domain of $L^{(n), \{r\}}$ , and

\begin{align*}L^{(n), \{l\}} [f^n h] =\lambda \mathbb{E} \int_{-\infty}^{\infty} d_y f_y^n ( h((y \vee A)+Z)-h(y) )+ \int_{-\infty}^{\infty} d_y f_y^n \int_y^{\infty} d_\xi f_{\xi}^n (h(\xi)-h(y)),\end{align*}

where the expectation is over the distribution of the random jump size Z. We also formally define the ‘limit’ $L^{\{l\}}$ of $L^{(n), \{l\}}$ (for pairs (f,A) with $f \in \mathcal{M}$ ) as

\begin{align*}L^{\{l\}} [f^n h] =\lambda \mathbb{E} \int_{-\infty}^{\infty} d_y f_y ( h((y \vee 0)+Z)-h(y) )+ \int_{-\infty}^{\infty} d_y f_y \int_y^{\infty} d_\xi f_{\xi} (h(\xi)-h(y)).\end{align*}

Theorem 4. Let $v>0$ and $A_0$ be fixed, and recall that the boundary is given by $A=A_0 +vt$ . Suppose $f^n(0) \stackrel{w}{\rightarrow} f(0)$ , where $\{f^n(0)\}$ is a deterministic sequence of elements $f^n(0) \in \mathcal{M}^{(n)}$ , and $f(0) \in \mathcal{M}$ . Then $f^n({\cdot}) \Rightarrow f({\cdot})$ in $D([0,\infty), \mathcal{M})$ , where $f({\cdot}) \in D([0,\infty), \mathcal{M})$ is deterministic and uniquely determined by f(0). Moreover, $f({\cdot})$ is a continuous element of $D([0,\infty), \mathcal{M})$ which satisfies

(5.5) \begin{equation}f(t) h - f(0) h - \int_0^t L^{\{l\}} f(s) h ds =0, \,\,\forall h\in \mathcal{C}_b, \,\forall t\ge 0.\end{equation}

This trajectory $f({\cdot})$ is called the mean-field limit (MFL) with initial state f(0). Furthermore, MFL $f({\cdot})$ is the unique MFM with initial state f(0); the MFM here is defined as in Definition 1 except that (5.1) is replaced by

(5.6) \begin{equation}\frac{\partial}{\partial t} f_x(t) = -\lambda f_x(t) - f_x(t) (1-f_x(t)), \,x \le A,\end{equation}

(5.7) \begin{equation}\frac{\partial}{\partial t} f_x(t) = -\lambda f_A(t) (1-J(x-A)) -\lambda \int_{A}^x d_y f_y(t) (1-J(x-y)) - f_x(t) (1-f_x(t)), \,x > A.\end{equation}

The proof of Theorem 4 is in Section 15.

6. MFLs that are traveling waves

An MFL (MFM) $f({\cdot})$ for our original system is a traveling wave if $f_x(t) =\phi_{x-vt}$ for some speed $v> 0$ and some $\phi = (\phi_x, x \in \mathbb{R}) \in \mathcal{M}$ , in which case we call $\phi$ a traveling-wave shape (TWS). Substituting $f_x(t) =\phi_{x-vt}$ into (5.1), we can easily see that a TWS, if it exists, must satisfy

(6.1) \begin{equation}v \phi^{\prime}_x = \lambda \int_{-\infty}^x \phi^{\prime}_y (1-J(x-y)) dy + \phi_x (1-\phi_x)\end{equation}

for each x, and in fact the derivative $\phi^{\prime}_x$ must be continuous in x.

Since a traveling wave is an MFL (which is a limit of particle system dynamics), we see that if a TWS with speed v exists, then necessarily $v \ge \lambda$ .

In the special case when $\lambda=0$ , the integro-differential equation (6.1) becomes the logistic differential equation

(6.2) \begin{equation}v \psi^{\prime}_x = \psi_x (1-\psi_x).\end{equation}

In this case, a unique (up to a shift) TWS does exist for every speed $v>0$ . This follows, for example, from [Reference Manita and Shcherbakov22, Proposition 5.1], but of course it is well known: the relevant solution to (6.2) is

(6.3) \begin{equation}\psi_x = \frac{1}{1+ e^{-(1/v)(x+c)}}, \,x\in \mathbb{R}, \,\,\mbox{where $v>0$ and $c\in \mathbb{R}$ are parameters}.\end{equation}

(There is no contradiction here with the fact that for any finite n, $\lambda=0$ implies that the steady-state speed $v_n=0$ . Traveling waves are MFLs, which means that we first take the $n\to\infty$ limit and then look at the time-evolution of the limit. So, if the time interval [0,T] is fixed, n is very large, and the initial state $f^n(0)$ is close to $\psi$ in (6.3), then the evolution of $f_x^n(t)$ in [0,T] is close to the traveling wave $\psi_{x-vt}$ moving at speed v. This does not contradict the fact that, eventually, all of the particles will assemble at the location of the leading particle and will stop moving, i.e. $v_n=0$ .)

In the special case of exponential jump size distribution, $J(x) =1- e^{-x}$ , for any $\lambda\ge 0$ , the integro-differential equation (6.1) becomes an ODE

(6.4) \begin{equation}v \phi''_x = (1+\lambda-v-2\phi_x) \phi^{\prime}_x + \phi_x (1-\phi_x).\end{equation}

Indeed, in this case (6.1) can be written as

\begin{align*}v \phi^{\prime}_x e^x = \lambda \int_{-\infty}^x \phi^{\prime}_y e^y dy + \phi_x (1-\phi_x) e^x,\end{align*}

from which, after differentiating in x, we obtain (6.4).

We will also consider MFLs that are traveling waves for the modified systems with moving right and left boundaries; $v>0$ is the speed of the boundary. Clearly, in a system with a boundary moving at speed v, the speed of any traveling wave is also v. For both systems, for a given boundary speed $v>0$ , an MFL $f({\cdot})$ is a traveling wave (of speed v) if $f_x(t) =\phi_{x-vt}$ for some $\phi = (\phi_x, x \in \mathbb{R}) \in \mathcal{M}$ , where $\phi$ is called a TWS for the corresponding system; without loss of generality we can assume that the initial location of the boundary is 0.

Substituting $f_x(t) =\phi_{x-vt}$ into (5.1), we see that a TWS $\phi$ for the system with moving right boundary, if it exists, must be continuous with $\phi_0=1$ and must satisfy (6.1) for each $x<0$ ; in fact, the derivative $\phi^{\prime}_x$ must be continuous in $({-}\infty,0]$ (if at $x=0$ we consider the left derivative). In the special case of $J(x) =1-e^{-x}$ , (6.1) reduces to (6.4).

Substituting $f_x(t) =\phi_{x-vt}$ into (5.7), we obtain that a TWS $\phi$ for the system with moving left boundary, if it exists, must be continuous for $x\in [0,\infty)$ and must satisfy

(6.5) \begin{equation}v \phi^{\prime}_x = \lambda \phi_0 (1-J(x)) + \lambda \int_{0+}^x \phi^{\prime}_y (1-J(x-y)) dy + \phi_x (1-\phi_x)\end{equation}

for each $x>0$ ; in fact, the derivative $\phi^{\prime}_x$ must be continuous in $[0,\infty)$ (if at $x=0$ we consider the right derivative). Note that the initial condition $v \phi^{\prime}_0 = \lambda \phi_0 + \phi_0(1-\phi)$ must hold. In the special case of $J(x) =1-e^{-x}$ , (6.5) reduces to (6.4).

7. Definition of the critical speed $v_{**}$ via an MSSP

In this section we formally define the critical value $v_{**}$ of the speed, which plays a central role in our results and analysis. Assume $\lambda>0$ . Denote by $L(s)= \int_0^\infty e^{-sx} dJ(x)$ the Laplace transform of the jump distribution $J({\cdot})$ . As a function of real s it is, of course, a convex non-negative non-increasing function. Denote by $\alpha$ the tail exponent of $J({\cdot})$ :

(7.1) \begin{equation}\alpha \doteq \liminf_{x\to\infty} \frac{-\log (1-J(x))}{x} = \sup \{\zeta \ge 0 \,|\, L({-}\zeta) < \infty\} \in [0,\infty].\end{equation}

Suppose $\alpha>0$ (with the case $\alpha=\infty$ allowed). Then $L(s) <+\infty$ for $s> - \alpha$ , $L(s) = +\infty$ for $s < - \alpha$ , and $L(s) \uparrow L({-}\alpha)$ as $s \downarrow -\alpha$ ; $L({-}\alpha)$ may be finite or infinite, but it is necessarily infinite when $\alpha=\infty$ .

The following heuristic argument, leading to (7.3), is analogous to the one in, e.g., [Reference Brunet and Derrida6], where it is applied to the KPP equation (1.1). This heuristic argument serves as a motivation for the rigorous definition of the speed value $v_{**}$ . Assume that a TWS $\phi = (\phi_x, \,x\in \mathbb{R})$ with speed $v>0$ exists, and that the front tail of $\phi$ decays exponentially: $1-\phi_x \sim e^{-\zeta x}$ as $x\to\infty$ , where $\zeta>0$ . Then, when x is large, Equation (6.1) ‘becomes’

(7.2) \begin{equation}v \zeta e^{-\zeta x} = \lambda \int_{-\infty}^x \zeta e^{-\zeta y} \bar J(x-y) dy + e^{-\zeta x}.\end{equation}

In (7.2),

\begin{align*}\int_{-\infty}^x e^{-\zeta y} \bar J(x-y) dy = e^{-\zeta x} \int_0^\infty e^{\zeta \xi} \bar J(\xi) d\xi= \frac{1}{\zeta} e^{-\zeta x} [L({-}\zeta)-1],\end{align*}

so we obtain the following dependence of the speed v on $\zeta>0$ :

(7.3) \begin{equation}v(\zeta) = \frac{1}{\zeta} [\lambda L({-}\zeta) - \lambda +1].\end{equation}

The relation (7.3), viewed formally, is a starting point for the following rigorous definitions and observations. The dependence $v(\zeta)$ is convex, because $(1/\zeta) [L({-}\zeta)-1]$ is the Laplace transform of function $\bar J({\cdot})$ at the point $s=-\zeta$ . Note also that $v(\zeta)\to +\infty$ as $\zeta\downarrow 0$ . It is easy to see that the minimum value $v_{**}$ of $v(\zeta)$ is attained at the unique positive finite value

(7.4) \begin{equation}\zeta_{**} \doteq \mathop{\mathrm{arg\,min}}\limits_{\zeta>0} v(\zeta) \in (0,\alpha], \,\,\mbox{so that}\,\, v_{**} \doteq v(\zeta_{**}) = \min_{\zeta>0} v(\zeta).\end{equation}

Because $\zeta_{**} \in (0,\alpha]$ and $L({-}\zeta) > 1$ for $\zeta>0$ , note from (7.3) that

(7.5) \begin{equation}v_{**} > 1/\alpha.\end{equation}

If $\zeta_{**} < \alpha$ , then it necessarily solves the equation

(7.6) \begin{equation}\lambda \zeta_{**} L'({-}\zeta_{**}) + \lambda L({-}\zeta_{**}) -\lambda +1 = 0;\end{equation}

and if $L({-}\alpha) =\infty$ , then necessarily $\zeta_{**} < \alpha$ . Note for future reference that the inverse function to $v(\zeta), \,0<\zeta \le \zeta_{**},$ is a continuous strictly decreasing convex function $\zeta(v), v_{**} \le v < \infty,$ with $\zeta(v) \downarrow 0$ as $v\uparrow\infty$ .

In the special case $\bar J(x) = e^{-x}$ , we have $\alpha=1$ , (7.3) becomes

(7.7) \begin{equation}v(\zeta) = \frac{\lambda}{1-\zeta} + \frac{1}{\zeta},\end{equation}

and then

\begin{align*}\zeta_{**} = \zeta_* \doteq \frac{1}{1+\sqrt{\lambda}}, \,\,v_{**} = v_* \doteq (1+\sqrt{\lambda})^2 .\end{align*}

The inverse function $\zeta(v), v_{*} \le v < \infty,$ of the function $v(\zeta), \,0<\zeta \le \zeta_{*},$ in this case is

(7.8) \begin{equation}\zeta(v) = \frac{(1+v-\lambda) - \sqrt{(1+v-\lambda)^2 - 4v}}{2v}.\end{equation}

Recall that the above definition of $v_{**}$ assumes $\alpha>0$ . In view of (7.5), we adopt the convention that $v_{**} = +\infty$ when $\alpha=0$ . All results in this paper involving the critical speed $v_{**}$ are valid for the case when $\alpha=0$ and $v_{**} = +\infty$ . They follow easily from the corresponding results for the case when $\alpha > 0$ and $v_{**} < \infty$ , thanks to monotonicity (Section 4). Specifically, if $\alpha=0$ , then we can compare the actual process to an auxiliary one with the jump size distribution slightly changed so that it is stochastically smaller, with an arbitrarily small positive tail exponent $\alpha$ . The auxiliary process, which is stochastically dominated by the actual one, can have an arbitrarily large value of $v_{**}$ . Thus, the corresponding results for the actual process hold with $v_{**}=\infty$ .

8. Limit of steady-state speeds and the associated BRW

8.1. McKean-type characterization of the original system’s BMFL

Theorem 5 (presented in this section) and its proof are analogous to the corresponding result and its proof in [Reference McKean23], for the system in which the independent movement of each particle is a Brownian motion.

Denote by $W(t), \,t \ge 0,$ a single-particle independent-jump (Markov) process, starting at some point y at initial time $t=0$ . Specifically, W(t) is the location at time t of a particle that starts at 0 and makes independent and identically distributed jumps, distributed as $J({\cdot})$ , at time points of a Poisson process of rate $\lambda>0$ .

Consider the following process, which we will call the associated (with our particle system) branching random walk (BRW). The process starts with a single particle, located initially at 0. Its location $W_1(t)$ evolves as a single-particle independent-jump process. Any particle present in the system generates a new particle in the same location (i.e. splits into two) according to a unit-rate Poisson process. Each newly created particle continues as a single-particle independent-jump process. We label particles in the order of their creation, so that at time t their locations are $W_1(t), \ldots, W_{N(t)}(t)$ , where N(t) is the total number of particles. Let $D(t) \doteq \max_i W_i(t)$ be the location of the rightmost particle of the associated BRW.

Theorem 5. The associated BRW is such that

\begin{align*}\mathbb{P}\{D(t) \le x\} = f_x(t),\end{align*}

where f(t) is the BMFL.

The proof of Theorem 5 is in Section 16.

8.2. The value $v_{**}$ as the limiting average speed of the leading particle of the associated BRW

The following proposition can be obtained from a general result on the average speed of progress of a leading particle of a BRW in discrete time, namely Theorem 3 in [Reference Biggins3].

Proposition 1. Consider the associated BRW, starting with one particle. Let D(t) be the location of the leading particle at time t. Then

(8.1) \begin{equation}D(t)/t \to v_{**}, \,\,\, with\ probability\ one.\end{equation}

Indeed, the process can viewed as follows. Each particle already present in the process ‘waits’ for an independent, exponentially distributed time with mean $1/(1+\lambda)$ , and then either jumps forward (with probability $\lambda/(1+\lambda)$ ) or splits into two particles at the same location (with probability $1/(1+\lambda)$ ). Suppose we discretize time, with the time step being some small $\delta>0$ . We obtain a process ‘slower’ than the original one if we assume that the time $\tau$ a particle ‘waits’ until the next event is geometric, with $\mathbb{P}\{\tau = j\delta\} = \varepsilon (1-\varepsilon)^{j-1}$ , $j=1,2,\ldots$ , $\varepsilon=1-\exp({-}(1+\lambda)\delta)$ . The particles present at time $n\delta$ represent the nth generation of the particles. The process is such that, after each time step, a particle has either one descendant in the same location (with probability $1-\varepsilon$ ), or two descendants in the same location (with probability $\varepsilon/(1+\lambda)$ ), or one descendant located at the current particle location + Z. Theorem 3 in [Reference Biggins3] can be applied to this discrete-time process to obtain the leading particle’s average speed of progress, which is a lower bound on the speed of progress for the original process. To obtain a discrete-time process which is ‘faster’ than the original one, we assume that the time $\tau$ a particle ‘waits’ until the next event is geometric, with $\mathbb{P}\{\tau = j\delta\} = \varepsilon (1-\varepsilon)^{j}$ , $j=0,1,2,\ldots$ . (A subtlety here is that $\tau=0$ with non-zero probability, so that the distribution of the set of descendants after one time step, i.e. after time $\delta$ , has to account for that.) Applying Theorem 3 in [Reference Biggins3] to this discrete-time process, we obtain an upper bound on the leading particle’s average speed of progress for the original process. Letting $\delta\downarrow 0$ , we can check that both bounds converge to exactly $v_{**}$ .

9. TWSs in the case of exponentially distributed independent jumps

Theorem 7. Consider the special case $J(x) =1-e^{-x}$ and recall the notation $v_* = (1+\sqrt{\lambda})^2$ . Assume $\lambda>0$ . Then the following hold:

(i) A TWS $\phi$ for the original system exists if and only if $v \ge v_*$ . If $v>v_*$ , then

(9.1) \begin{equation}- \lim_{x\to\infty} [\log (1-\phi_x)]/x = \zeta(v),\end{equation}

where $\zeta(v)$ is defined in (7.8).

(ii) For any $v > v_*$ there exists a TWS $\phi$ for the system with moving left boundary at speed v, and it is such that (9.1) holds.

(iii) For any $v < v_*$ there exists a unique TWS $\phi$ for the system with moving right boundary at speed v.

The proof of Theorem 7 is in Section 18.

10. MFL speed results

The following is a natural and standard notion of the average speed of a given MFL. We say that the average speed of an MFL f(t) with a given initial state $f(0) \in \mathcal{M}$ is lower-bounded (upper-bounded) by v if, for any quantile $\nu \in (0,1)$ of f(t), its average speed of progress is lower-bounded (upper-bounded) by v; that is,

\begin{align*}\liminf_{t\to\infty}\, (\limsup_{t\to\infty}) \, \frac{1}{t}[q_\nu(f(t)) - q_\nu(f(0))] \le \,(\ge)\, v.\end{align*}

An equivalent definition of a lower (upper) speed bound v is that for any $\varepsilon >0$ ,

\begin{align*}\lim_{t\to\infty} f_{(v-\varepsilon) t}(t) =0 \,\, \left( \lim_{t\to\infty} f_{(v+\varepsilon) t}(t) =1 \right).\end{align*}

The speed v is the average speed of an MFL if it is both a lower and an upper bound.

Clearly an MFL’s average speed depends on the initial state, and it can be arbitrarily large. (This follows from the existence of a traveling wave with $\lambda=0$ and arbitrarily large speed v.)

As a corollary of Theorem 5 and Proposition 1 we obtain the following result.

Proposition 2 in turn is used to prove (in Section 19) the following theorem.

In cases when one can establish the existence of traveling waves, including traveling waves with moving boundaries—as we did in the case of exponential jump sizes—those traveling waves can be used as lower and/or upper bounds for MFLs to obtain more precise MFL speed bounds. Specifically, we can use the following fact, which is a corollary of Lemma 1.

The proof of Theorem 9 is in Section 20. This proof does not rely on Proposition 2, which in turn relies on the connection to the associated BRW and its properties, as described in Theorem 5 and Proposition 1. It relies only on Theorem 7 and Proposition 3. (This is why we include the statement (i) in Theorem 9, even though it is a special case of Proposition 2 and Theorem 8. We give an alternative proof of (i).)

The proof of Theorem 9 shows how MFL speed bounds can be obtained directly from the existence and properties of traveling waves. In our case these are given by Theorem 7. However, the proof is quite generic. For example, some of the MFL speed bounds for the KPP model under certain initial conditions, as obtained in [Reference Freidlin9, Reference Freidlin10, Reference Larson16, Reference Veretennikov27] via direct analysis or via Feynmann–Kac representations and large-deviations analysis, can be instead obtained in essentially the same way as our Theorem 9. Indeed, given the traveling-wave existence results for the KPP model [Reference Kolmogorov, Petrovskii and Piscounov15], it is straightforward to obtain an analogue of our Theorem 7, namely the existence of traveling waves with moving left and right boundaries for speeds larger and smaller, respectively, than the critical speed. Then an analogue of our Theorem 9 follows, with essentially the same proof.

11. Conjectures, simulation results, and discussion

11.2. Simulations and discussion

Before presenting our simulation results, we note that if the objective is to maximize the steady-state speed of progress of a large-scale system, then our Theorem 6 allows one to easily optimize the trade-off between $\lambda$ and $\mu$ , if these quantities are chosen subject to some constraint(s). For example, one may want to optimize the trade-off between the levels of effort allocated to self-propulsion and synchronization, if both require the consumption of a common limited resource. Our simulations show good accuracy of the optimal setting based on Theorem 6.

Tables 1 and 2 show steady-state speeds $v_n$ , obtained by simulation of a system with $n=10000$ particles, under exponentially and uniformly (in [0, 2) distributed jump sizes, respectively. (For each simulation run, there are $400 \times n$ attempted jumps in total, i.e. 400 per particle. This corresponds to simulation time $400 \times n / (\lambda+\mu)$ . The first half of the simulation run is a warm-up.) The tables also show the values of $v_{**}$ . (For the case of exponential jump sizes, $v_{**}= v_*=(\sqrt{\lambda} + \sqrt{\mu})^2$ .)

Table 1. Exponential jump sizes, $n=10000$ , $2\lambda+\mu=1$ . Steady-state speed $v_n$ (simulated) and $v_{**}=\lim_n v_n$ . (For exponential jump sizes, $v_{**} = (\sqrt{\lambda} + \sqrt{\mu})^2$ .) The pair $(\lambda_{opt}, \mu_{opt})$ maximizes $v_{**}$ .

Table 2. Uniform[0, 2] jump sizes, $n=10000$ , $2\lambda+\mu=1$ . Steady-state speed $v_n$ (simulated) and $v_{**}=\lim_n v_n$ . The pair $(\lambda_{opt}, \mu_{opt})$ maximizes $v_{**}$ .

The results for exponential jump sizes are in Table 1. We see that the actual steady-state speed $v_n$ stays below $v_{**}$ , as we know it should. We also observe that, even for a relatively large number of particles (10000), the difference $v_{10000} - v_{**}$ is still significant. This is not unexpected, because the convergence $v_n \uparrow v_{**}$ in other contexts is known to be rather slow [Reference Brunet and Derrida5, Reference Brunet and Derrida6]. Formally showing that this is true for our system as well would be an interesting problem for further research. The results for the uniform jump sizes in Table 2 show the same patterns.

Suppose now that independent jumps and synchronizations by a particle consume some common resource (say, computing power or energy). To be specific, suppose the maximum rate at which the resource may be consumed by a particle is normalized to 1, and the amounts of the resource consumed by one independent jump and one synchronization are $a>0$ and $b>0$ , respectively. Suppose the aim is to maximize the speed of the particle system. Assume the number of particles is large enough so that $v_n \approx v_{**}$ . Then we obtain the optimization problem

(11.1) \begin{equation}\max v_{**} \,\,\mbox{subject to} \,\, a \lambda + b \mu = 1.\end{equation}

In the case of exponentially distributed jumps, we have $v_{**}= (\sqrt{\lambda} + \sqrt{\mu})^2$ , and the problem (11.1) has a simple explicit solution

\begin{align*}\lambda_{opt} = \frac{1}{a+a^2/b}, \,\,\mu_{opt} = \frac{1}{b+b^2/a}.\end{align*}

For a general jump size distribution, the optimal solution $(\lambda_{opt},\mu_{opt})$ can easily be found numerically using (7.3) and (7.4).

In Tables 1 and 2, for the cases of exponential and uniform jumps, we vary $(\lambda,\mu)$ while satisfying the constraint $a\lambda+b\mu=1$ with $a=2, b=1$ . In both cases we show $(\lambda_{opt},\mu_{opt})$ . We see that, even though the values of $v_n$ for $n=10000$ are not necessarily very close to their limiting values $v_{**}$ ‘yet’ (because this n is not large enough), the parameter setting $(\lambda_{opt},\mu_{opt})$ obtained from (11.1) essentially maximizes $v_n$ for this n. Thus, the setting $(\lambda_{opt},\mu_{opt})$ , which is easily computable, gives a practical rule of thumb for optimizing the trade-off between self-propulsion and synchronization in large-scale systems.

12. Proof of Theorem 2

12.1. Characterization of distributional limit

Theorem 10. Suppose $f^n(0) \stackrel{w}{\rightarrow} f(0)$ , as in Theorem 2. Then the sequence of processes $f^n({\cdot})$ is tight in the space $D([0,\infty), \mathcal{M}^B)$ , and any sub-sequential distributional limit $f({\cdot})$ is such that, with probability one, the trajectory $f({\cdot})$ is a continuous element of $D([0,\infty), \mathcal{M}^B)$ which satisfies (5.3).

Proof. The proof is easily obtained using the steps given in the proof of Theorem 3 in [Reference Stolyar25] (for a different particle system), namely Theorems 4–7 and Corollary 8 in [Reference Stolyar25]. (In turn, the proof of Theorem 3 in [Reference Stolyar25] closely follows the development in [Reference Balázs, Rácz and Tóth1].) The only substantial difference is the verification of the condition (i) of [Reference Perkins24], which in our notation has the following form: for any $T>0$ and $\varepsilon>0$ , there exists $K>0$ such that

(12.1) \begin{equation}\sup_n \mathbb{P} \left(\sup_{t\le T} [f^n_{-K}(t) + 1- f^n_{K}(t)] >\varepsilon \right) < \varepsilon.\end{equation}

However, for the process under consideration, with fixed right boundary at the point B, (12.1) is trivial—it suffices to choose K large enough so that $K > B$ and $f^n_{-K}(0) \le \varepsilon/2$ uniformly in n.

12.2. Equivalent characterization of solutions to (5.3) as MFMs

Theorem 11. For any initial condition $f(0) \in \mathcal{M}^B$ , a trajectory $f({\cdot}) \in D([0,\infty), \mathcal{M}^B)$ satisfies (5.3) if and only if it is an MFM.

Proof. This proof is very similar to the proof of Theorem 10 in [Reference Stolyar25].

‘Only if’ statement: We only need to consider points $x<B$ . Let $h=(h(u), \,u\in \mathbb{R})$ be the step function $h(u) = {\mathbf{I}}(u \le x)$ , jumping from 1 to 0 at the point x. Let $h_\varepsilon$ , $\varepsilon>0$ , be a continuous approximation of h that is linearly decreasing from 1 to 0 in $[x,x+\varepsilon]$ . We see that

\begin{align*}L^B [f(t) h_\varepsilon] \to L^B [f(t) h], \,\,\forall t.\end{align*}

Indeed, $|L^B [f(t) h_\varepsilon] - L^B [f(t) h]|$ is upper-bounded by $(\lambda+1)$ times the probability that a random jump—either independent of the size Z or thanks to synchronization—of a particle randomly located according to the distribution f(t) is such that the jump either originates or lands in $(x,x+\varepsilon)$ ; this probability vanishes as $\varepsilon\downarrow 0$ . Also, since both $L^B [f(t) h_\varepsilon]$ and $L^B [f(t) h_\varepsilon]$ are within $[-(\lambda+1),0]$ , for all t and $\varepsilon$ , we have a universal bound $|L [f(t) h_\varepsilon] - L [f(t) h]| \le \lambda+1$ . Then, for any fixed t, by taking the $\varepsilon\downarrow 0$ limit in $f(t) h_\varepsilon - f(0) h_\varepsilon - \int_0^t L^B [f(s) h_\varepsilon] ds =0$ , we obtain

\begin{align*}f(t) h - f(0) h - \int_0^t L^B [f(s) h] ds =0.\end{align*}

This means that $f(t) h = f_x(t)$ is absolutely continuous in t, with the derivative equal to $(\partial/\partial t) f_x(t)=L^B [f(t) h]$ almost everywhere (with respect to the Lebesgue measure) in t. In particular this implies that for any fixed y, the difference $f_y(t)-f_{y-}(t)$ is continuous in t (in fact, Lipschitz); this, in turn, means that, possible discontinuity points y of $f_y(t)$ ‘cannot move’ at time t. We can then conclude that, for any x, the derivative $(\partial/\partial t) f_x(t)=L^B [f(t) h]$ is in fact continuous in t. Therefore, $(\partial/\partial t) f_x(t)=L^B [f(t) h]$ at every t. It remains to observe that $L^B [f(t) h]$ is exactly the right-hand side of (5.1).

‘If’ statement: The definition of an MFM $f({\cdot})$ implies that (5.3) holds for the above-defined step function h for any $x<B$ . Then we have (5.3) for any h, which is piecewise constant with a finite number of pieces; and the set of such functions h is tight within the space of test functions $h \in \mathcal{C}_b$ , equipped with the uniform metric. Thus (5.3) holds for any $h \in \mathcal{C}_b$ .

12.3. MFM uniqueness

Theorem 12. For any initial condition $f(0) \in \mathcal{M}^B$ , an MFM $f({\cdot}) \in D([0,\infty), \mathcal{M}^B)$ is unique.

Proof. The existence of an MFM for a given initial state f(0) follows from Theorems 10 and 11. For any MFM $f({\cdot})$ we know that $f_x(t)$ , as a function of t, is Lipschitz, uniformly in x. Consider two MFM trajectories, $f({\cdot})$ and $g({\cdot})$ , with the same initial state $f(0)=g(0)$ . It is easy to check that for any x and any $t>0$

(12.2) \begin{equation}\left| \frac{d}{d t} g_x(t) - \frac{d}{d t} f_x(t) \right| \le C \| g(t) - f(t) \|\end{equation}

for a fixed constant $C>0$ (for example, for $C=2\lambda+1$ ). From (12.2) it is easy to see that

(12.3) \begin{equation} \frac{d}{d t} \left\| g(t) - f(t) \right\| \le C \| g(t) - f(t) \|,\end{equation}

as long as $\| g(t) - f(t) \| >0$ . But then, by Gronwall’s inequality, $\| g(t) - f(t)\| \equiv 0$ . This proves the uniqueness.

13. Proof of Theorem 1

We now generalize Theorem 2 to Theorem 1. Recall that the evolution of $f_x^n({\cdot}), x \in ({-}\infty, B),$ is independent of the behavior of particles located in $[B, \infty)$ . Theorem 2 holds for every B for the projection $f^{n,B}({\cdot}) \in D([0,\infty), \mathcal{M}^B)$ of our original process $f^{n}({\cdot})$ , and we have convergence to the corresponding unique continuous MFL (MFM) $f^B({\cdot})$ . By uniqueness, these MFLs for different B must be consistent; that is, for any $B_1 < B_2$ , any $x< B_1$ , and any t we have $f_x^{B_1}(t)=f_x^{B_2}(t)$ . So we can formally define a trajectory f(t) by $f_x(t) \doteq \lim_{B\uparrow \infty} f_x^B(t)$ . Let us show that this trajectory is the MFL satisfying the statement of Theorem 1.

We first show that $f(t) \in \mathcal{M}$ for any t, i.e. $f_\infty(t) = 1$ . Recall that, for any x, $(d/dt)f_x(t)$ is non-positive, bounded, and Lipschitz continuous in t (uniformly in x). Then $\varepsilon(t) \doteq 1-f_\infty(t)$ is non-decreasing and continuous, starting from 0 at time 0. Moreover, $(d/dt)\varepsilon(t)$ is bounded and Lipschitz continuous. Consider a sequence $c \uparrow \infty$ and the corresponding (non-decreasing) sequence of space-rescaled versions of f(t),

\begin{align*}f^{(c)}_x(t) = f_{cx}(t).\end{align*}

Clearly, $f^{(c)}(t)$ is the limiting trajectory of the process with rescaled initial state and rescaled jumps; and for any t the limit of $ f_{cx}(t)$ , as a function of x, is constant, equal to $1-\varepsilon(t)$ . Consider any fixed $x>0$ and $t>0$ . As $c\to\infty$ , $1- f^{(c)}_x(t) \to \varepsilon(t)$ and $(d/dt)f^{(c)}_x(t) \to -\varepsilon(t)(1-\varepsilon(t))$ . For fixed $t < s$ ,

\begin{align*}\varepsilon(s) - \varepsilon(t) = \lim_{c\to\infty} [-f^{(c)}_x(s) + f^{(c)}_x(t)]= - \lim_{c\to\infty} \int_t^s (d/d\xi)f^{(c)}_x(\xi) d\xi = \int_t^s \varepsilon(\xi)(1-\varepsilon(\xi)) d\xi.\end{align*}

We conclude that $\varepsilon(t)$ , as a function of t, must be a solution to the logistic equation $\varepsilon' = \varepsilon(1-\varepsilon)$ with initial condition $\varepsilon(0)=0$ . Therefore, $\varepsilon(t) \equiv 0$ , and then $f_\infty(t)\equiv 0$ , i.e. $f(t) \in \mathcal{M}$ .

Now, because we know that $f(t) \in \mathcal{M}$ for any t, $f({\cdot})$ is clearly an MFM: it satisfies the conditions of Definition 1 for any x (because we can always choose $B >x$ ). The condition (5.2) holds for any $h \in \mathcal{C}_b$ , because we can choose B large enough so that h(x) is constant in $[B,\infty)$ ; for such h, the operators $L f(t) h = L^B f(t) h$ , and then (5.2) follows from (5.3).

It remains to show the convergence $f^n({\cdot}) \Rightarrow f({\cdot})$ . Fix any $T>0$ , any $\varepsilon>0$ , and a sufficiently large $B=B(T)>0$ so that $1-f_B(T) \le \varepsilon$ . For the modified system with the right boundary at B, we have convergence of $f^{n,B}({\cdot}) \in D([0,T], \mathcal{M}^B)$ to a continuous $f^{B}({\cdot}) \in D([0,T], \mathcal{M}^B)$ . But since $\varepsilon>0$ can be arbitrarily small, we also have convergence of the original process $f^{n}({\cdot}) \in D([0,T], \mathcal{M})$ to a continuous $f({\cdot}) \in D([0,T], \mathcal{M})$ . This is true for arbitrary $T>0$ . Therefore, we also have convergence of the original process $f^{n}({\cdot}) \in D([0,\infty), \mathcal{M})$ to a continuous $f({\cdot}) \in D([0,\infty), \mathcal{M})$ .

14. Proof of Theorem 3

The proof is essentially the same as that of Theorems 1 and 2, with straightforward adjustments. The condition (12.1) for this process is automatic.

We only comment on the proof of MFM uniqueness. Note that for any MFM $f({\cdot})$ , for each x, the derivative $(d/dt) f_x(t)$ can have a discontinuity at time t such that $B=x$ (i.e. when the moving boundary ‘hits’ the point x). However, this does not affect the uniqueness proof in Section 12.3—we just need to note that (12.2) and (12.3) hold for almost any $t>0$ with respect to the Lebesgue measure.

15. Proof of Theorem 4

This proof is also essentially the same as that of Theorems 1 and 2, with straightforward adjustments.

Again, we only comment on the proof of MFM uniqueness. An MFM for this system is such that, for a given x, the derivative $(d/dt) f_x(t)$ may have a discontinuity (a jump down) at any time t such that the jump distribution has an atom at $x-A$ . If we consider two MFM trajectories, $f({\cdot})$ and $g({\cdot})$ , with the same initial state $f(0)=g(0)$ , as in the uniqueness proof in Section 12.3, then for each x, both $(d/dt) f_x(t)$ and $(d/dt) g_x(t)$ will have the same countable set of potential discontinuities in time. We also have that, at any t, $f_x(t)$ and $g_x(t)$ , as functions of x, have the same set of discontinuity points. Given these facts, we see that the uniqueness proof in Section 12.3 applies—again, we just need to note that (12.2) and (12.3) hold for almost any $t>0$ with respect to the Lebesgue measure.

16. Proof of Theorem 5

For the single-particle independent-jump (Markov) process $W({\cdot})$ , denote by $P^t(x,H)$ its transition function and by $P^t$ its transition operator,

\begin{align*}P^t g_x = \int_x^\infty P^t(x,dy) g_y.\end{align*}

The generator of this process is

(16.1) \begin{equation}B g_x = \lambda \left[ \int_0^\infty g_{x+\eta} d\bar J(\eta) - g_x \right].\end{equation}

Any function g such that $1-g \in \mathcal{M}$ (i.e. a complementary distribution function) is within the domain of the generator B, and for such functions, integration by parts in (16.1) gives

(16.2) \begin{equation}B g_x = \lambda \int_x^\infty \bar J(\eta-x) d g_\eta.\end{equation}

(The integral in (16.2) includes a possible atom of measure $g_\eta$ at $\eta=x$ . If g is continuous, this subtlety is irrelevant.)

Let us fix a function g such that $1-g \in \mathcal{M}$ , and consider the function

(16.3) \begin{equation}u_x(t) = \mathbb{E} [g_{x+W_1(t)} \cdot \ldots \cdot g_{x+W_{N(t)}(t)}].\end{equation}

Clearly, $u_x(0)=g_x$ . Note that if $g_x = {\mathbf{I}}\{x \le 0\} = \chi_{-x}$ , then $u_x(t) = \mathbb{P}\{\max_i W_i(t) \le 0\}$ . Considering two cases—depending on whether or not the first split of the first particles occurs in [0,t]—we obtain

\begin{align*}u_x(t) = e^{-t} P^t g_x + \int_0^t & e^{-s} P^{s} u^2_x(t-s) ds = e^{-t} P^t g_x + \int_0^t e^{-(t-\xi)} P^{t-\xi} u^2_x(\xi) d\xi,\\[5pt]& e^t u_x(t) = P^t g_x + \int_0^t e^\xi P^{t-\xi} u^2_x(\xi) d\xi.\end{align*}

Differentiating in t, we obtain

\begin{align*}e^t u_x(t) + e^t \frac{\partial}{\partial t}u_x(t) =B P^t g_x+ e^t P^0 u^2_x(t)+ \int_0^t e^\xi P^{t-\xi} B u^2_x(\xi) d\xi;\end{align*}

then

\begin{align*}\frac{\partial}{\partial t}u_x(t) = B u_x(t) + u^2_x(t) - u_x(t),\end{align*}

and finally, recalling (16.2), we have

(16.4) \begin{equation}\frac{\partial}{\partial t}u_x(t) = \lambda \int_x^\infty \bar J(\eta-x) d_\eta u_\eta(t) + u^2_x(t) - u_x(t).\end{equation}

(It is easy to see from the definition of $u_x(t)$ that, for any $t>0$ , it is continuous in x. Therefore the subtlety mentioned immediately after (16.2) is irrelevant.)

Now let $g_x = {\mathbf{I}}\{x \le 0\}$ . Recall that $u_x(t) = \mathbb{P}\{\max_i W_i(t) \le 0\}$ , and then

\begin{align*}u_{-x}(t) = \mathbb{P}\{\max_i W_i(t) \le x\}.\end{align*}

Substituting $f_x(t)=u_{-x}(t)$ in (16.4), we obtain exactly (5.1) with $f(0)=\chi$ .

17. Proof of Theorem 6

The monotonicity, $v_n \le v_{n+1}$ , has already been observed in (4.1).

Let us show that $v_n \le v_{**}$ for all n. The proof of this fact is the same as the proof of the analogous upper bound in Theorem 2.3(4) in [Reference Groisman, Jonckheere and Martínez13], except for our process we need to use Proposition 1, instead of the analogous fact for the process in [Reference Groisman, Jonckheere and Martínez13] (which follows directly from the results of [Reference Kolmogorov, Petrovskii and Piscounov15, Reference McKean23]). Here is the proof, for completeness. Consider the system with a finite number n of particles. Consider another, artificial, system starting with the same initial state with n particles, but such that each initial particle generates its own independent associated BRW as defined above. Clearly, the two systems can be coupled so that the artificial system dominates the actual one in the sense of $\preceq_l$ , and in particular the location of its rightmost particle, M(t), is always to the right of location of the rightmost particle, $M_n(t)$ , in the actual system. But Proposition 1 holds for each of the n independent associated BRWs. We infer that $\limsup_{t\to\infty} M_n(t)/t \le \limsup_{t\to\infty} M(t)/t = v_{**}$ with probability one, which implies $v_n \le v_{**}$ .

It remains to show that $\liminf_n v_n \ge v_{**}.$ Consider an arbitrary initial state of the process with n particles. Suppose without loss of generality that the leading particle is initially at 0, and let $D_n(t)$ be the location of the leading particle at time t. Consider a modified process such that all particles except the leading one are initially placed at $-\infty$ . By monotonicity, the location $\tilde D_n(t)$ of the leading particle in the modified system is stochastically dominated by $D_n(t)$ . The modified process is such that, in particular, at some points in time some particles located at $-\infty$ will jump forward to join particles in $[0,\infty)$ ; let us call the particles located in $[0,\infty)$ regular. Let us now fix a finite time interval [0,T], and consider the limit of the process of regular particles in this time interval as $n\to\infty$ . It is easy to construct a coupling such that, with probability one, the process of regular particles (of the modified system) converges to an associated BRW. (Indeed, when n is large, the rate at which each regular particle is being joined by a particle from $-\infty$ is close to 1. We omit the details, which are straightforward.) By Proposition 1, if T is sufficiently large, then the average speed of advance of the leading particle of the BRW in [0,T] is close to $v_{**}$ . (This is if $\alpha>0$ and then $v_{**} < \infty$ . If $\alpha=0$ and then $v_{**} = \infty$ , ‘close to $v_{**}$ ’ means ‘arbitrarily large’.) Consequently, for all large n, the average speed of advance of $\tilde D_n(t)$ is close to $v_{**}$ . This implies (we skip straightforward $\varepsilon$ – $\delta$ formalities) that $\liminf_n v_n \ge v_{**}$ .

18. Existence and properties of traveling waves in the case of exponentially distributed jump sizes; proof of Theorem 7

In this section we consider the case of exponential jump size distribution, $J(x) =1- e^{-x}$ , and develop results leading to the proof of Theorem 7.

Recall that in this case a TWS $\phi=(\phi_x)$ must satisfy the ODE (6.4):

(18.1) \begin{equation}v \phi'' = (1+\lambda-v-2\phi) \phi' + \phi(1-\phi).\end{equation}

A function $\phi=(\phi_x)$ is a TWS for our original system if and only if it is a proper solution of the ODE (18.1)—that is, if and only if $\phi_x \in [0,1]$ for all $x \in \mathbb{R}$ , (18.1) holds for all $x \in \mathbb{R}$ , $\lim_{x \downarrow -\infty} \phi_x = 0$ , and $\lim_{x \uparrow \infty} \phi_x = 1$ .

It will be convenient to consider the ODE (18.1) in terms of the first-order ODE in phase space $(\phi, z=\phi')$ :

(18.2) \begin{equation}\phi' = z, \,\,z' = -z + \frac{1+\lambda-2\phi}{v}z + \frac{\phi}{v}(1-\phi).\end{equation}

In terms of (18.2), a proper solution is a solution which starts at the point (0,0), ends at the point (1, 0), and stays within the strip $\phi \in [0,1]$ .

The following are immediate observations from (18.2). Solutions to (18.2) (not necessarily proper solutions) have the following properties: a solution trajectory cannot hit the $\phi$ -axis for $\phi\in (0,1)$ ; given any two solution trajectories, either they coincide for $\phi \in [0,1]$ or one strictly dominates another for $\phi \in (0,1)$ .

18.1. Lower bound on speed v, for any proper solution

Consider the linearization of (18.2) at the point (1, 0). If we set $\nu=1-\phi$ , the linearization is

(18.3) \begin{equation}\nu' = -z, \,\,z' = -z + \frac{\lambda-1}{v}z + \frac{1}{v}\nu.\end{equation}

The eigenvalues of this linear system satisfy the characteristic equation

(18.4) \begin{equation}v \zeta^2 + (1+v -\lambda) \zeta + 1 =0.\end{equation}

We know (see Section 6) that for a proper solution to exist, we must have $v\ge \lambda$ . Given that, $-(1+v-\lambda) < 0$ , and it is easy to check that the condition $(1+v-\lambda)^2 -4v \ge 0$ is equivalent to the condition $v \ge v_* \doteq (1+\sqrt{\lambda})^2$ . We see that both eigenvalues of (18.3) (roots of (18.4)) have negative real parts; the eigenvalues are two (different) adjoint complex numbers if and only if $v<(1+\sqrt{\lambda})^2$ , and they are both real otherwise. We then observe that when $v<(1+\sqrt{\lambda})^2$ , a proper solution does not exist, because any solution with endpoint at (1, 0) converges to (1, 0) by oscillating around it (as illustrated in Figure 1) and thus must ‘hit’ the line $\phi=1$ at a point strictly above the point (1, 0).

Figure 1. Vector field $(\phi',z')$ in the vicinity of (1, 0); $\lambda=4$ , $v=7 < v_*=9$ .

We conclude that if a proper solution of (18.2) exists, then necessarily $v\ge v_* = (1+\sqrt{\lambda})^2.$

18.2. Proof of Theorem 7(i)

To complete the proof of Theorem 9(i), we need to show that a a proper solution of (18.2) exists for any $v \ge v_*$ . We will prove that it exists for $v> v_*$ , and then will obtain the existence for $v=v_*$ by continuity.

18.2.1. Existence, uniqueness, and continuity of solutions to (18.2)

Here we consider solutions to (18.2) (not necessarily proper solutions) within the domain $\{0\le \phi \le 1, \,z\ge 0\}$ . Consider a fixed parameter $v \ge v_*$ and an initial condition $(\phi_0,z_0)$ for a solution to (18.2). (The initial condition can be viewed as a point which a solution trajectory must contain, because the trajectories satisfying (18.2) can be considered in both the forward and the reverse direction.) From the standard theory of ODEs, we see that the solution for a given triple $(\phi_0,z_0,v)$ exists and is unique as long as $(\phi_0,z_0)\ne (0,0)$ and $(\phi_0,z_0)\ne (1, 0)$ . Moreover, the solution is continuous in $(\phi_0,z_0,v)$ outside those two points. The following lemma shows that the existence, uniqueness, and continuity hold for the initial condition $(\phi_0,z_0)= (0,0)$ as well, if we consider non-trivial solutions.

Lemma 2. For a fixed $\lambda >0$ and any speed $v\ge v_*$ , there exists a unique non-trivial solution $(\phi, z(\phi))$ such that $z(\phi)>0$ and $(\phi, z(\phi)) \to (0,0)$ as $\phi\downarrow 0$ . This solution is such that $dz/d\phi = \gamma$ at the initial $\phi=0$ , where

(18.5) \begin{equation}\gamma=\frac{-(v-1-\lambda) + \sqrt{(v-1-\lambda)^2 +4v}}{2v} >0.\end{equation}

Moreover, the solution is continuous with respect to $(\phi_0,z_0,v)$ .

Proof of Lemma 2. Recall that $v \ge v_* > 1+ \lambda$ . Then, for the linearization of (18.2) at the point (0,0), we always have two different real eigenvalues, one positive and one negative: $-\gamma_2 < 0 < \gamma$ , where $\gamma$ is given by (18.5). The corresponding eigenvectors are $(1,\gamma)$ and $(1,-\gamma_2)$ ; see Figure 2.

Figure 2. Vector field $(\phi',z')$ in the vicinity of (0,0); $\lambda=4$ , $v=10$ , $v > v_*$ .

The proof of existence is constructive. Consider any sequence of solutions, each starting at a point $(\phi^{(k)}_0, z^{(k)}_0) \ge (0,0)$ and corresponding to a speed $v^{(k)} \ge v_*$ ; the sequence is such that, as $n\to\infty$ , $(\phi^{(k)}_0, z^{(k)}_0) \to (0,0)$ , $v^{(k)} \to v$ , and $(\phi^{(k)}_0, z^{(k)}_0) \ne (0,0)$ for all k. We view these solutions as functions $z^{(k)}=z^{(k)}(\phi)$ ; this is possible because $\phi'=z>0$ for $\phi \in (\phi_0, 1)$ . Then any sub-sequential limit of this sequence of solutions must be a solution $z=z(\phi)$ (with speed parameter v) starting from (0,0). Also, given the structure of the derivative vector field around (0,0), it is easy to see that any such solution must satisfy $dz/d\phi = \gamma$ at $\phi=0$ . (Indeed, for $c>0$ , denote by $M_c = \{z=c\phi, \,\phi \ge 0\}$ the ray from the point (0,0) in the direction (1,c). For a small fixed $\varepsilon>0$ , denote by W the closed cone between the rays $M_{\gamma-\varepsilon}$ and $M_{\gamma+\varepsilon}$ . If we look at the solution in the reverse direction, as it approaches the point (0,0), it cannot be outside W arbitrarily close to (0,0)—otherwise it will hit the z-axis strictly above (0,0).)

It remains to show the uniqueness of a solution for a given parameter v. (Then the continuity in $(\phi_0,z_0,v)$ will automatically follow.) Consider any two solutions to (18.2), viewed as functions $z=z(\phi)$ and $\hat z= \hat z(\phi)$ . It is easy to see from (18.2) that

\begin{align*}\frac{d}{d\phi} [\hat z - z] = \frac{\phi(1-\phi)}{v} \left[ \frac{1}{\hat z} - \frac{1}{z} \right]= - \frac{\phi(1-\phi)}{v\hat z z} [\hat z - z].\end{align*}

Thus, the difference $\hat z(\phi) - z(\phi)$ cannot increase in $\phi$ , which implies uniqueness.

18.2.2. Existence of a proper solution for $v \ge v_*$

Lemma 3. A proper solution of (18.2) exists for any $v>v_*$ . (Furthermore, by continuity of solutions in v, it exists for $v=v_*$ as well.)

Proof of Lemma 3. Consider $C>1$ (to be determined later). Consider a point $(\phi,z)$ , with $\phi\in (0,1)$ and $z\ge 0$ . Let us compare the derivative vector $(\phi',z')$ , given by (18.2), to the tangent vector to the parabola $h(\phi) = C \phi(1-\phi)/v$ at the point $\phi$ , with the $\phi$ -component equal to $\phi'$ ; the tangent vector is then $(\phi', \phi' C(1-2\phi)/v)$ . The difference between the z-components of these two vectors is

\[z' - \phi' C(1-2\phi)/v =-\left(\frac{v-\lambda+(C-1)(1-2\phi)}{v}-\frac{1}{C}\right)z-\frac{1}{C}(z-h(\phi)).\]

If $(\phi,z)=(\phi, h(\phi))$ , i.e. the point $(\phi,z)$ is on the parabola, this difference is

\begin{align*}-\left(\frac{v-\lambda+(C-1)(1-2\phi)}{v}-\frac{1}{C}\right)z.\end{align*}

Let us consider the expression in the bracket (a linear function of $\phi$ ). If we can prove that it is positive for $0\leq \phi\leq 1$ , then the trajectories of our dynamic system will traverse the parabola $h=C\phi(1-\phi)/v$ entering the domain between that parabola and $\phi$ -axis. (See Figure 3.) Let us check that this is the case. The coefficient in front of $\phi$ is negative, so it suffices to check the expression is positive at $\phi=1$ , i.e.

\[\frac{v-\lambda-(C-1)}{v}-\frac{1}{C}>0, \mbox{ i.e., }\frac{v-\lambda+1}{v}>\frac{C}{v}+\frac{1}{C}.\]

The condition $v>(1+\sqrt{\lambda})^2$ is equivalent to $\sqrt{v}>1+\sqrt{\lambda}$ , then to $(\sqrt{v}-1)^2>\lambda$ , then to $v-\lambda+1>2\sqrt{v}$ . If we take $C=\sqrt{v}$ , this gives exactly the last display.

Figure 3. Vector field $(\phi',z')$ , for points inside the domain between the parabolas $z = C \phi(1-\phi)/v$ and $z=\phi(1-\phi)/v$ ; here $C=\sqrt{v}$ , $\lambda=4$ , $v=10$ , $v > v_*$ .

Now, consider the solution $z(\phi)$ to (18.2) with initial condition (0,0). The derivative $dz/d\phi$ at $\phi=0$ is $\gamma$ (in (18.5)), while the derivative of $h(\phi)$ (with $C=\sqrt{v}$ ) is $1/\sqrt{v}$ . For $\lambda>0$ and $v \ge v_*$ , the inequality $1/\sqrt{v} > \gamma$ can be verified by simple algebra. This means that, for sufficiently small $\phi>0$ , the solution $z(\phi)$ is ‘under’ the parabola $h(\phi)$ . But, as shown above, it cannot traverse $h(\phi)$ . We also know that the solution $z(\phi)$ cannot hit the $\phi$ -axis while $\phi\in (0,1)$ . We conclude that $z(\phi) \le h(\phi)$ for all $\phi \le 1$ , and $z(1)=0$ . In particular, $z(\phi)$ is a proper solution. (In fact, one can show that the trajectories also are transversal to the parabola $z=\phi(1-\phi)/v$ .)

18.2.3. Derivative of the proper solution at the point (1, 0)

To complete the proof of of Theorem 7(i), it remains to show that (9.1) holds when $v > v_*$ . If $v>v_*$ , the linearization of our dynamic system at the point (1, 0) has two strictly negative eigenvalues, $-\zeta_1 > -\zeta_2,$ , being roots of (18.4), so that

\begin{align*}\zeta_1 = \frac{(1+v-\lambda) - \sqrt{(1+v-\lambda)^2 - 4v}}{2v}, \,\,\,\zeta_2 = \frac{(1+v-\lambda) + \sqrt{(1+v-\lambda)^2 - 4v}}{2v}.\end{align*}

Since the solution $z=z(\phi)$ for speed v is dominated by the parabola $\phi(1-\phi)/\sqrt{v}$ , we see that in the neighborhood of (1, 0) it is dominated by the line $z=(1-\phi)/\sqrt{v}$ . It is easy to see that $1/\sqrt{v} < \zeta_2$ . (Indeed, the stronger inequality $1/\sqrt{v} < (1+v-\lambda)/(2v)$ is equivalent to $v > v_*$ .) Therefore, in the neighborhood of (1, 0) the solution $z=z(\phi)$ is separated from the eigendirection $z=\zeta_2 (1-\phi)$ by line $z=(1-\phi)/\sqrt{v}$ . But then the solution $z=z(\phi)$ must approach the point (1, 0) along the eigendirection $z=\zeta_1 (1-\phi)$ , that is

\begin{align*}\left. \frac{dz}{d\phi} \right|_{\phi=1} = - \zeta_1.\end{align*}

It remains to notice that $\zeta_1$ is exactly equal to $\zeta(v)$ in (7.8).

19. Proof of Theorem 8

Consider a fixed MFL $f({\cdot})$ . Let us fix an arbitrarily small $\nu>0$ . Without loss of generality, assume that $f_0(0) \ge \nu$ and $f_{0-}(0) \le \nu$ . Recall that the trajectory $f({\cdot})$ is the unique distributional limit of a sequence of processes $f^n({\cdot})$ with deterministic initial states such that $f^n(0) \stackrel{w}{\rightarrow} f(0)$ . We can and will choose a sequence of initial states such that, for all n, $\nu n$ of the leftmost particles are located in $({-}\infty,0]$ (let us call them ‘left particles’) and then the remaining $(1-\nu)n$ particles are located in $[0,\infty)$ (let us call them ‘right particles’).

Now, for each n, consider a lower-bounding process $\hat f^n({\cdot})$ constructed as follows. The initial state of $\hat f^n({\cdot})$ is such that the $\nu n$ ‘left’ particles’ locations are the same as in the original process, i.e. $\hat f_x^n(0) = f_x^n(0)$ for $x< 0$ , while the $(1-\nu) n$ ‘right’ particles are located exactly at 0. The evolution of $\hat f^n({\cdot})$ is such that the ‘left’ particles are ‘frozen’—they do not jump at all; the ‘right’ particles jump as usual, making both independent and synchronization jumps. Clearly, $\hat f^n({\cdot})$ and $f^n({\cdot})$ can be coupled so that $\hat f^n(t) \preceq f^n(t)$ at all times; then $\hat f(t) \preceq f(t)$ for the corresponding MFLs. Now, observe that the evolution of $\hat f^n({\cdot})$ is such that the left particles can simply be ignored, and all of the right particles are initially at 0. Next observe that the process $\hat f^n({\cdot})$ is such that the evolution of the set of $(1-\nu)n$ right particles can be viewed as that of a stand-alone original system, except the synchronization rate of each particle is $(1-\nu)$ instead of 1. (Recall that in the system under consideration we have $\mu=1$ , without loss of generality.) Therefore, the MFL $\hat f({\cdot})$ has the following form: $\hat f_x(t) =f_x(0)(t)$ for $x<0$ , and

(19.1) \begin{equation}\hat f_x(t) =\nu + (1-\nu) \tilde f_x(t), \,\,\mbox{for}\, x>0,\end{equation}

where $\tilde f({\cdot})$ is the BMFL of the system with independent jump rate $\lambda$ and synchronization rate $1-\nu$ . For a sufficiently small $\nu>0$ , the speed $v^{\prime}_{**}$ of $\tilde f({\cdot})$ is arbitrarily close to $v_{**}$ . (In particular, if $\alpha=0$ , then $v^{\prime}_{**}=v_{**}=\infty$ .) We conclude that any quantile $\beta > \nu$ of $f({\cdot})$ advances at average speed at least $v^{\prime}_{**}$ . Since $\nu>0$ can be chosen arbitrarily small, any quantile $\beta > 0$ of $f({\cdot})$ advances at average speed at least $v_{**}$ .

20. Proof of Theorem 10

(i) Obviously, this property is a special case of the more general properties given in Proposition 2 and Theorem 8. However, we now give a simple alternative proof, which does not rely on the connection to and properties of the associated BRW, described in Theorem 7 and Proposition 1. The proof relies only on Theorem 8 and monotonicity.

By Theorem 7(ii)–(iii), the BMFL can be lower- and upper-dominated by the following traveling waves: a traveling wave with moving upper boundary with speed v’ slightly smaller that $v_*$ , and a traveling wave with moving lower boundary with speed v” slightly larger that $v_*$ . Since v’ and v” can be arbitrarily close to $v_*$ , the average speed of BMFL is $v_*$ . To prove that $v_*$ is a lower bound on any MFL’s average speed, the proof of Theorem 10 applies.

(ii) By Theorem 7(ii) there exists a TWS $\psi$ with moving left boundary and speed v’ slightly greater than $v(\zeta)$ . The right tail exponent of $\psi$ is slightly smaller than $\zeta$ . We can always consider a version of this $\psi$ , shifted sufficiently far to the right, so that $f(0) \preceq \psi$ . By monotonicity, MFL $f({\cdot})$ is dominated by the traveling wave (with moving left boundary) with speed v’, starting from $\psi$ . Therefore, the average speed of the MFL is upper-bounded by v’, which can chosen arbitrarily close to $v(\zeta)$ .

(iii) When $\zeta = \zeta_*$ , the claimed property follows from (i). Therefore, it suffices to consider the case $\zeta < \zeta_*$ . Fix a small $\nu>0$ . If $\nu$ is small enough, by Theorem 7(iii) there exists a TWS $\psi$ for the system with synchronization rate $\mu=1-\nu$ , with speed v’ slightly smaller than $v(\zeta)$ , and with the right tail exponent of $\psi$ slightly larger than $\zeta$ . Consider now the function $\tilde \psi = \nu + (1-\nu) \psi$ , and consider the version of it shifted sufficiently far to the left, so that $\tilde \psi \preceq f(0)$ . This $\tilde \psi$ can be viewed as the initial state of an MFL for a system where the $\nu n$ leftmost particles are located at $-\infty$ and are frozen (i.e., never make any jumps); then the corresponding MFL is a traveling wave with speed v’, dominated by $f({\cdot})$ . We see that any quantile $\beta > \nu$ of f(t) advances at average speed at least v’. Since $\nu$ can be arbitrarily small and v’ can be arbitrarily close to $v(\zeta)$ , we obtain the claim.