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Uniqueness and Longtime Behavior of the Completely Positively Correlated Symbiotic Branching Model

Eran Avneri, Leonid Mytnik

TL;DR

This work resolves the long-standing open problem of weak uniqueness for the completely positively correlated symbiotic branching model ($\rho=1$) by establishing a novel duality with the parabolic Anderson model. The authors reformulate the SBM via a pair of martingale problems and prove a Laplace-transform duality that implies uniqueness in law for the associated martingale problem. Leveraging this duality, they analyze the longtime behavior: under integrable initial data, one population dies out and the surviving population retains mass equal to the initial difference; under mean-defined initial data, the dying-out population converges to zero in probability while the surviving population converges to a constant multiple of the spatial mean, demonstrating non-coexistence in both global and local senses. The results provide a powerful tool for studying coexistence in SBM and suggest potential extensions to other correlation regimes and discrete-space analogs.

Abstract

The symbiotic branching model in $\mathbb{R}$ describes the behavior of two branching populations migrating in space $\mathbb{R}$ in terms of a corresponding system of stochastic partial differential equations. The system is parametrized with a correlation parameter $ρ$, which takes values in $[-1,1]$ and governs the correlation between the branching mechanisms of the two populations. While existence and uniqueness for this system were established for $ρ\in [-1,1)$, weak uniqueness for the completely positively correlated case of $ρ= 1$ has been an open problem. In this paper, we resolve this problem, establishing weak uniqueness for the corresponding system of stochastic partial differential equations. The proof uses a new duality between the symbiotic branching model and the well-known parabolic Anderson model. Furthermore, we use this duality to investigate the long-term behavior of the completely positively correlated symbiotic branching model. We show that, under suitable initial conditions, after a long time, one of the populations dies out. We treat the case of integrable initial conditions and the case of bounded non-integrable initial conditions with well-defined mean.

Uniqueness and Longtime Behavior of the Completely Positively Correlated Symbiotic Branching Model

TL;DR

This work resolves the long-standing open problem of weak uniqueness for the completely positively correlated symbiotic branching model () by establishing a novel duality with the parabolic Anderson model. The authors reformulate the SBM via a pair of martingale problems and prove a Laplace-transform duality that implies uniqueness in law for the associated martingale problem. Leveraging this duality, they analyze the longtime behavior: under integrable initial data, one population dies out and the surviving population retains mass equal to the initial difference; under mean-defined initial data, the dying-out population converges to zero in probability while the surviving population converges to a constant multiple of the spatial mean, demonstrating non-coexistence in both global and local senses. The results provide a powerful tool for studying coexistence in SBM and suggest potential extensions to other correlation regimes and discrete-space analogs.

Abstract

The symbiotic branching model in describes the behavior of two branching populations migrating in space in terms of a corresponding system of stochastic partial differential equations. The system is parametrized with a correlation parameter , which takes values in and governs the correlation between the branching mechanisms of the two populations. While existence and uniqueness for this system were established for , weak uniqueness for the completely positively correlated case of has been an open problem. In this paper, we resolve this problem, establishing weak uniqueness for the corresponding system of stochastic partial differential equations. The proof uses a new duality between the symbiotic branching model and the well-known parabolic Anderson model. Furthermore, we use this duality to investigate the long-term behavior of the completely positively correlated symbiotic branching model. We show that, under suitable initial conditions, after a long time, one of the populations dies out. We treat the case of integrable initial conditions and the case of bounded non-integrable initial conditions with well-defined mean.

Paper Structure

This paper contains 8 sections, 16 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Symbiotic Branching Model
  3. Preliminaries: notation and spaces
  4. Main Results
  5. Proofs
  6. Proof of Theorem \ref{['prop: duality']}
  7. Longtime behavior. Proofs of Theorems \ref{['main_res: global_longtime']}, \ref{['main_res: local_longtime']}.
  8. Acknowledgments

Key Result

Theorem 2.3

Let $\mu \in \mathscr{P}(C_{\text{tem}}^+ \times C_{\text{tem}}^+)$ with compact support. Let $(U^1_t, U^2_t)_{t \geq 0}$ be a solution of the martingale problem $(MP)_U$ with initial distribution $\mu$ and sample paths in $C_{C_{\text{tem}}^+ \times C_{\text{tem}}^+}[0,\infty)$. Let $\phi \in C_{\t where $X_t = U^1_t + U^2_t, Y_t = U^1_t - U^2_t = S_t(U^1_0 - U^2_0).$

Theorems & Definitions (37)

  • Remark 2.1
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Lemma 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 27 more