On the subcritical self-catalytic branching Brownian motions

TL;DR

This work analyzes subcritical self-catalytic branching Brownian motions (SBBM) with infinite initial particles by establishing a robust duality framework linking the particle system to a stochastic reaction-diffusion SPDE and its mean-field counterpart. The authors prove existence of SBBM in the infinite-particle regime, show non-explosion, and derive a complete CDI theory: observables localized to a region remain finite iff the region intersects the initial trace in a bounded way, with CDI rates governed by the mean-field PDE solution v_t and the catalytic coupling Ψ'(0+). The CDI behavior is universal with respect to the precise offspring laws, depending only on the initial trace and the mean-field parameters, and extends the duality machinery to infinite configurations via careful martingale and PDE techniques. The results connect SBBM dynamics to stochastic FKPP-type SPDEs, offer a pathway to understanding KPZ-type universality in critical regimes, and lay groundwork for further exploration of CDI beyond subcritical catalytic interactions.

Abstract

The self-catalytic branching Brownian motions (SBBM) are extensions of the classical one-dimensional branching Brownian motions by incorporating pairwise branchings catalyzed by the intersection local times of the particle pairs. These processes naturally arise as the moment duals of certain reaction-diffusion equations perturbed by multiplicative space-time white noise. For the subcritical case of the catalytic branching mechanism, we construct the SBBM allowing an infinite number of initial particles. Additionally, we establish the coming down from infinity (CDI) property for these systems and characterize their CDI rates.

Theorems & Definitions (90)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4