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Branching Particle Systems with Mutually Catalytic Interactions

Alexandra Jamchi Fugenfirov, Leonid Mytnik

TL;DR

This work analyzes a two-type mutually catalytic branching model on $\mathbb{Z}^d$, where each population’s branching rate at a site is proportional to the other’s local mass. It establishes existence and uniqueness of the interacting particle system, derives moment formulas, and demonstrates a finite-system renormalization scheme that yields a diffusion limit. The authors prove a coexistence dichotomy: coexistence is possible in transient settings (e.g., $d\ge 3$) for finite masses and not in recurrent ones ($d\le 2$), and they show that the renormalized finite-system totals converge to a diffusion $(X_t,Y_t)$ with $dX_t=\sqrt{\gamma\sigma^2 X_tY_t}\,dW^1_t$, $dY_t=\sqrt{\gamma\sigma^2 X_tY_t}\,dW^2_t$ and initial $(X_0,Y_0)=(\theta_1,\theta_2)$. A key methodological contribution is the approximating duality approach, which enables convergence proofs in the absence of self-duality. The results extend the finite-system perspective of Dawson–Perkins diffusions to a particle-based setting and provide a framework for analyzing limit theorems for interacting multi-type spatial systems.

Abstract

We study a continuous time Mutually Catalytic Branching model on the $\mathbb{Z}^{d}$. The model describes the behavior of two different populations of particles, performing random walk on the lattice in the presence of branching, that is, each particle dies at a certain rate and is replaced by a random number of offspring. The branching rate of a particle in one population is proportional to the number of particles of another population at the same site. We study the long time behavior for this model, in particular, coexistence and non-coexistence of two populations in the long run. Finally, we construct a sequence of renormalized processes and use duality techniques to investigate its limiting behavior.

Branching Particle Systems with Mutually Catalytic Interactions

TL;DR

This work analyzes a two-type mutually catalytic branching model on , where each population’s branching rate at a site is proportional to the other’s local mass. It establishes existence and uniqueness of the interacting particle system, derives moment formulas, and demonstrates a finite-system renormalization scheme that yields a diffusion limit. The authors prove a coexistence dichotomy: coexistence is possible in transient settings (e.g., ) for finite masses and not in recurrent ones (), and they show that the renormalized finite-system totals converge to a diffusion with , and initial . A key methodological contribution is the approximating duality approach, which enables convergence proofs in the absence of self-duality. The results extend the finite-system perspective of Dawson–Perkins diffusions to a particle-based setting and provide a framework for analyzing limit theorems for interacting multi-type spatial systems.

Abstract

We study a continuous time Mutually Catalytic Branching model on the . The model describes the behavior of two different populations of particles, performing random walk on the lattice in the presence of branching, that is, each particle dies at a certain rate and is replaced by a random number of offspring. The branching rate of a particle in one population is proportional to the number of particles of another population at the same site. We study the long time behavior for this model, in particular, coexistence and non-coexistence of two populations in the long run. Finally, we construct a sequence of renormalized processes and use duality techniques to investigate its limiting behavior.
Paper Structure (14 sections, 26 theorems, 90 equations)

This paper contains 14 sections, 26 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Paper overview
  4. Our Model and Main Results
  5. Description of the Model
  6. Main Results
  7. Existence and Uniqueness. Proof of Theorem \ref{['lem:Lemma-1']}
  8. Proof of Theorem \ref{['thm:Coexistance-finite-1']}.
  9. Proof of Theorem \ref{['thm:Coexistance-finite-1']}(a) $-$ Transient case
  10. Proof of Theorem \ref{['thm:Coexistance-finite-1']}(b) $-$ Recurrent case
  11. Moment computations for $S=\Lambda_{n}$
  12. Proof of Theorem \ref{['thm:MainResult-1']}
  13. Proof of Proposition \ref{['prop:Laplace-Fourier']}
  14. Proof of Proposition \ref{['lem:residue']}

Key Result

Theorem 1.1

(Theorem 1(a) in CDG04) Let $d\geq3$, and let $Q$ be a generator of a simple random walk on $\mathbb{Z}^{d}$. Assume that Then where $\left(X_{t},Y_{t}\right)_{t\geq0}$ is the unique weak solution for the following system of stochastic differential equations with initial conditions $(X_{0},Y_{0})=\bar{\theta}=(\theta_1,\theta_2)$, where $w^{1},\, w^{2}$ are two independent standard Brownian mot

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Lemma 3.1
  • ...and 22 more