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Two-Sided Free Boundary Problems Arising From Branching-Selection Particle Systems

This paper introduces the $(N,p)$ -BBM, a two-sided branching-selection particle system that generalizes the classical $N$ -BBM, and establishes its convergence to a deterministic hydrodynamic limit described by a two-sided free boundary problem, proving the existence, regularity, and asymptotic velocity of this limit as the particle number $N$ tends to infinity.

Original authors: Jacob Mercer

Published 2026-04-24

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CC BY 4.0

Original authors: Jacob Mercer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded dance floor where everyone is moving randomly, bumping into each other, and occasionally splitting into two people. This is the basic idea behind a Branching Particle System.

Now, imagine this dance floor has a strict rule: The room can only hold $N$ people. Every time someone splits into two, the total number of people becomes $N+1$ . To fix this, the "bouncer" must kick one person out immediately.

In the classic version of this game (called the N-BBM), the bouncer is very picky: they always kick out the person standing furthest to the left. This simulates "survival of the fittest" in a one-sided way: the population constantly pushes to the right, evolving to become "fitter" (moving further right) over time.

Jacob Mercer's paper introduces a new, more complex version of this game called the (N, p)-BBM.

The New Game: A Two-Sided Bouncer

In this new version, the bouncer has a coin.

When a split happens, the bouncer flips a coin.
If it's Heads (probability $p$ ), they kick out the person furthest to the left.
If it's Tails (probability $1-p$ ), they kick out the person furthest to the right.

Why does this matter?

If $p=1$ , it's the old game (only left kicks). The crowd just runs away to the right.
If $p=0$ , it's the mirror image (only right kicks). The crowd runs away to the left.
If $p=0.5$ (a fair coin), the crowd is being squeezed from both sides. The leftmost person might get kicked, or the rightmost person might get kicked. The crowd stays in a "Goldilocks zone"—not too wide, not too narrow, but constantly shifting.

The Big Discovery: The "Fluid" Limit

The paper asks: What happens if we have a huge number of people ( $N$ goes to infinity)?

Instead of tracking individual dancers, the author looks at the density of the crowd as if it were a fluid or a cloud of smoke.

The Shape: In the old game, the cloud of people stretched out infinitely to the right, with a sharp edge on the left. In the new game, the cloud gets squeezed into a finite box. It has a moving left wall and a moving right wall.
The Movement: The whole box moves forward (or backward) at a specific speed.
The Math: The author proves that as the number of people gets huge, the chaotic random dancing settles down into a predictable, smooth pattern described by a Free Boundary Problem.
- Think of this like a balloon being inflated and pushed. The air inside (the people) spreads out, but the rubber skin (the boundaries) moves to keep the volume constant. The math describes exactly how that skin moves and how the air inside flows.

The "Speed" of Evolution

One of the most fascinating results is about speed.

In the old game (kicking only the left), the crowd moves at a specific speed determined by how fast they branch.
In the new game, the speed depends entirely on the coin flip probability $p$ $p$ .
- If you favor kicking the left ( $p > 0.5$ ), the crowd moves right.
- If you favor kicking the right ( $p < 0.5$ ), the crowd moves left.
- The author found a precise formula for this speed. It's like finding the exact speed limit for a car based on how much you press the gas vs. the brake.

The "Inverse" Mystery

To prove that this "fluid box" actually exists and behaves nicely, the author used a clever trick involving First Passage Problems.

Normal Problem: "If I drop a ball from here, how long until it hits the wall?"
Inverse Problem (used here): "I want the ball to hit the wall at a specific time with a specific probability. Where do I have to build the wall?"

The author showed that the moving walls of the crowd are exactly the walls you would need to build to make a random walker behave in a very specific way. This connection allowed him to prove that the crowd's shape is smooth and well-behaved, not jagged or chaotic.

Real-World Analogies

Why should we care about a bunch of math particles?

Evolution: Imagine a species where being too small is bad (you get eaten), but being too large is also bad (you can't find food). The population naturally evolves to stay in a middle range. The left wall represents "too small," and the right wall represents "too large." The paper models how this population stabilizes.
Fire: Imagine a flame spreading. Sometimes it burns out on the left, sometimes on the right. The "two-sided" nature of this model helps scientists understand how flames propagate through complex environments.
Traffic: Imagine a highway where cars are constantly merging (branching) and exiting. If you only let cars exit at the back, traffic jams move forward. If you let them exit at both ends, the traffic jam stays in a specific zone, moving at a steady pace.

Summary in a Nutshell

Jacob Mercer took a famous math model of evolution (where the "unfit" on the left are removed) and made it two-sided (removing the unfit on both ends).

He proved that:

With enough people, the chaos turns into a smooth, predictable wave.
This wave has a specific speed determined by the rules of the game.
The shape of this wave is a "moving box" with smooth walls, and we can calculate exactly how fast that box moves.

It's a beautiful piece of mathematics that connects random chaos (individual particles) to deterministic order (fluid waves), showing us how nature might balance itself when pressure comes from both sides.

1. Problem Statement and Motivation

The paper investigates the hydrodynamic limit of a generalized branching-selection particle system known as the $(N, p)$ -BBM (Branching Brownian Motion).

Context: The study builds upon the well-known $N$ -BBM model, where $N$ particles perform independent Brownian motions and branch at rate 1. In the standard $N$ -BBM, upon every branching event, the leftmost particle is deleted to maintain a constant population size $N$ . This models "survival of the fittest" where fitness corresponds to position on the real line.
The Generalization: The $(N, p)$ $(N, p)$ -BBM introduces a parameter $p \in (0, 1)$ $p \in (0, 1)$ . When a branching event occurs:
- With probability $p$ , the leftmost particle is deleted.
- With probability $1-p$ , the rightmost particle is deleted.
- The case $p=1$ recovers the standard $N$ -BBM.
Motivation: While the $N$ -BBM leads to a one-sided free boundary problem (a single moving boundary), the $(N, p)$ -BBM with $p \in (0, 1)$ is expected to generate a two-sided free boundary problem with two moving boundaries ( $L_t$ and $R_t$ ). This models scenarios where selection acts on both extremes of a trait distribution (e.g., predator-prey dynamics or flame propagation), rather than just one tail.

2. Methodology

The author employs a rigorous probabilistic and analytical approach combining coupling techniques, inverse first passage problems, and subadditive ergodic theory.

A. Coupling and Discretization

To handle the randomness of deletion times, the author constructs upper and lower bounding particle systems ( $X^{N,p,\delta,+}$ and $X^{N,p,\delta,-}$ ) where selection/deletion occurs only at discrete time steps $k\delta$ .

Coupling: These systems are coupled to the original $(N, p)$ -BBM such that $X^{N,p,\delta,-} \preceq X^{N,p} \preceq X^{N,p,\delta,+}$ at discrete times.
Hydrodynamic Limits of Bounds: The empirical distributions of these bounding systems are shown to converge to deterministic limits $u^{\delta,+}$ and $u^{\delta,-}$ as $N \to \infty$ . These limits are defined via operators involving Gaussian diffusion ( $G_\delta$ ) and mass-cutting operators ( $C_L, C_R$ ) that remove specific masses from the left and right.

B. Connection to Inverse First Passage Problems

A crucial theoretical bridge is established between the Free Boundary Problem (FBP) and Inverse First Passage Problems (IFPP).

The paper posits that the moving boundaries $L_t$ and $R_t$ of the limiting FBP correspond to the boundaries of a Brownian motion $W$ such that the first exit time $\tau = \inf\{t > 0 : W_t \notin (L_t, R_t)\}$ is exponentially distributed with rate 1.
Furthermore, the probability of exiting via the left boundary is $p$ , and via the right is $1-p$ .
By leveraging existing results on the existence and regularity of solutions to the asymmetric two-sided IFPP (specifically for exponential exit times), the author proves the existence and $C^\infty$ regularity of the free boundaries for the FBP.

C. Subadditive Ergodic Theorem

To establish the existence of an asymptotic velocity for the particle system, the author applies Kingman's Subadditive Ergodic Theorem. By defining a subadditive process based on the position of the rightmost (and leftmost) particles, it is proven that the system possesses a well-defined asymptotic velocity $v_N$ almost surely and in $L^1$ .

3. Key Contributions and Results

A. The Hydrodynamic Limit (Theorem 2.2)

The main result establishes that as $N \to \infty$ , the empirical distribution of the $(N, p)$ -BBM converges to the unique solution $u(x,t)$ of the following two-sided free boundary problem:

$\begin{cases} u_t = \frac{1}{2}u_{xx} + u & x \in (L_t, R_t), \, t > 0 \\ u(x, t) = 0 & x \notin (L_t, R_t) \\ u_x(L_t, t) = 2p, \quad u_x(R_t, t) = 2(p-1) & t > 0 \\ \int_{L_t}^{R_t} u(y, t) dy = 1 & t > 0 \\ u(x, 0) = \rho(x) \end{cases}$

Significance: This generalizes the one-sided FBP of the $N$ -BBM. The Neumann boundary conditions ( $u_x = 2p$ and $2(p-1)$ ) explicitly encode the selection probabilities $p$ and $1-p$ .
Uniqueness: The paper proves that this FBP has a unique solution with smooth ( $C^\infty$ ) boundaries.

B. Travelling Wave Solutions (Proposition 2.4)

The author derives an explicit unique travelling wave solution for the limiting FBP.

The solution takes the form $u(x,t) = w(x-ct)$ with boundaries $L_t = ct$ and $R_t = R_0 + ct$ .
The wave speed $c$ is uniquely determined by $p$ :
$c = \text{sgn}(p-1/2) \sqrt{\frac{2 \log^2(p/(1-p))}{\log^2(p/(1-p)) + \pi^2}}$
Unlike the $N$ -BBM (which has infinitely many travelling waves and selects the minimal speed $\sqrt{2}$ ), the $(N, p)$ -BBM selects a unique travelling wave speed determined by the asymmetry parameter $p$ .

C. Convergence of Boundaries and Speeds (Theorems 2.3 & 2.5)

Boundary Convergence: The leftmost ( $X^N_1$ ) and rightmost ( $X^N_N$ ) particles of the $(N, p)$ -BBM converge almost surely and in expectation to the free boundaries $L_t$ and $R_t$ of the limiting PDE.
Speed Convergence: The asymptotic velocity $v_N$ of the $(N, p)$ -BBM converges to the unique travelling wave speed $c$ of the limiting FBP as $N \to \infty$ .
$\lim_{N \to \infty} v_N = c$

4. Significance and Impact

Generalization of Selection Principles: The paper extends the "selection principle" (where particle systems select specific travelling waves) from one-sided to two-sided selection. It demonstrates that the specific asymmetry in deletion rules ( $p$ vs $1-p$ ) dictates a unique wave speed, resolving the non-uniqueness seen in the standard $N$ -BBM.
Mathematical Rigor: It provides a complete existence and regularity theory for a class of two-sided free boundary problems with Neumann conditions, linking them to inverse first passage problems.
Applications: The model offers a mathematical framework for evolutionary dynamics where selection pressures act on both extremes of a trait distribution (e.g., stabilizing selection or environmental constraints affecting both ends of a spectrum) and for physical phenomena like flame propagation with two moving fronts.
Technical Innovation: The use of discrete-time coupling to approximate continuous-time selection, combined with the probabilistic representation of the FBP via inverse first passage times, provides a robust toolkit for analyzing similar interacting particle systems.

In summary, Jacob Mercer successfully characterizes the macroscopic behavior of a generalized branching-selection system, proving its convergence to a unique, regular two-sided free boundary problem and explicitly calculating the resulting asymptotic velocity.