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Distribution of particles near the front in supercritical branching Brownian motion with compactly supported branching

Pratima Hebbar, Leonid Koralov

TL;DR

The paper analyzes the long-time near-front distribution of a $d$-dimensional supercritical branching Brownian motion with a compactly supported branching potential. It develops a recursive PDE framework for densities and higher-order moments, leveraging sharp parabolic PDE asymptotics in a compact potential and the principal eigenfunction $\\psi$ of the generator $\\mathcal{L}=\\tfrac{1}{2}\\Delta + v$ with top eigenvalue $\\lambda_0$. Central contributions include explicit moment formulas via recursive functions $G_k$ and their normalized limits $f^k$ and $f^k_{b,\\mathbf{u}}$, and the proof that the normalized near-front counts $\\eta^x(t,y(t))$ converge in distribution and in all moments to limits that depend on the asymptotic front offset. The results yield random variables $\\xi^x$ and $\\xi^x_{b,\\mathbf{u}}$ with those moments, extending previous one-dimensional and homogeneous-front findings to higher dimensions and to the compactly supported setting. Overall, the work clarifies the near-front population structure, shows absence of intermittency for the near-front limit, and demonstrates that the limiting distribution encodes the asymptotic location of the moving observation window.

Abstract

We investigate the long-time behavior of a $d-$dimensional supercritical branching Brownian motion with a compactly supported branching potential. It is known that, for $\mathbf{v}\in \mathbb{R}^d$, all the moments of the normalized number of particles in a bounded domain centered at $\mathbf{v} t$ converge, as $t \rightarrow \infty$, provided that $\|\mathbf{v}\|$ is strictly less than the asymptotic speed of the front. The limiting distribution does not depend on $\mathbf{v}$. Using sharp asymptotics for the solutions of parabolic PDEs with compact potential, we prove that the normalized number of particles in a bounded time-dependent domain located near the front converges in distribution and with all the moments. The limit, however, now depends on the asymptotic location of the domain.

Distribution of particles near the front in supercritical branching Brownian motion with compactly supported branching

TL;DR

The paper analyzes the long-time near-front distribution of a -dimensional supercritical branching Brownian motion with a compactly supported branching potential. It develops a recursive PDE framework for densities and higher-order moments, leveraging sharp parabolic PDE asymptotics in a compact potential and the principal eigenfunction of the generator with top eigenvalue . Central contributions include explicit moment formulas via recursive functions and their normalized limits and , and the proof that the normalized near-front counts converge in distribution and in all moments to limits that depend on the asymptotic front offset. The results yield random variables and with those moments, extending previous one-dimensional and homogeneous-front findings to higher dimensions and to the compactly supported setting. Overall, the work clarifies the near-front population structure, shows absence of intermittency for the near-front limit, and demonstrates that the limiting distribution encodes the asymptotic location of the moving observation window.

Abstract

We investigate the long-time behavior of a dimensional supercritical branching Brownian motion with a compactly supported branching potential. It is known that, for , all the moments of the normalized number of particles in a bounded domain centered at converge, as , provided that is strictly less than the asymptotic speed of the front. The limiting distribution does not depend on . Using sharp asymptotics for the solutions of parabolic PDEs with compact potential, we prove that the normalized number of particles in a bounded time-dependent domain located near the front converges in distribution and with all the moments. The limit, however, now depends on the asymptotic location of the domain.
Paper Structure (4 sections, 4 theorems, 89 equations)

This paper contains 4 sections, 4 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Higher order moments and correlation functions.
  4. Proof of Main Result

Key Result

Theorem 2.1

There exist random variables $\xi^x_{b,\mathbf{u}}$ and $\xi^x$ with moments which will be written out explicitly, such that, for each $k \geq 1$, where $y(t)$ is given by (yfr). The limit in (mainform1) is uniform in the location of the initial particle $x \in \Gamma$ for any compact set $\Gamma$.

Theorems & Definitions (9)

  • Theorem 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 3.1
  • Remark 3.2
  • Lemma 4.1
  • Lemma 4.2
  • proof : Proof of Theorem \ref{['mainresult']}