Linear and uniform in time bound for the binary branching model with Moran type interactions

TL;DR

The paper analyzes the binary branching model with Moran-type interactions (BBMMI) and its link to the Feynman-Kac semigroup, focusing on error control for approximating the semigroup by an interacting particle system. Building on the CHV2024 framework, it proves that under suitable regularity, the L2-distance between the normalized empirical measure and the FK semigroup can be bounded linearly in time, rather than exponentially, with bounds depending on the sequence $\alpha_t$ and the function $h$. It identifies conditions under which the linear-in-$T$ bound holds (e.g., $h$ bounded away from zero and summable or exponentially decaying $\alpha_t$), discusses uniform convergence via quasi-stationary distribution theory, and presents informative counterexamples. The Brownian motion with drift on a bounded $C^2$ domain is treated as a key application, where a coupling to jumping reflected Brownian motions yields an optimal $O(1/\sqrt{N_{\min}})$ bound, extending Fleming–Viot-type results to BBMMI.

Abstract

In this note, we recall the definition of the binary branching model with Moran type interactions (BBMMI) introduced in [8]. In this interacting particle system, particles evolve, reproduce and die independently and, with a probability that may depend on the configuration of the whole system, the death of a particle may trigger the reproduction of another particle, while a branching event may trigger the death of another particle. We recall its relation to the Feynman-Kac semigroup of the underlying Markov evolution and improve on the L 2 distance between their normalisations proved in [8], when additional regularity is assumed on the process.

Figures (1)

• Figure 1: A schematic representation of the $N_{min}-N_{max}$ dynamic with $N_{min}=3$ and $N_{max}=4$. The process starts with $N=4$ particles at time $0$. The first event is a killing, so that the number of particles goes down to $N=3= N_{min}$. The next event is a killing, so that the number of particles goes down to $2<N_{min}$, which triggers a resampling event: one of the $2$ remaining particles (chosen uniformly at random) is duplicated, and the number of particles goes back to $N=2+1=N_{min}$. The next event is a branching, so that the number of particles goes up to $N=4= N_{max}$. The subsequent event is a branching event, so that the number of particles goes up to $5>N_{max}$, which triggers a selection event, so that one of the $5$ particles (chosen uniformly at random) is removed from the system, and the number of particles goes back to $N=4=N_{max}$. The next event is a killing, so that the number of particles goes down to $3= N_{min}$, and so on.

Theorems & Definitions (12)

• Theorem 1
• Remark 2
• proof
• Remark 3
• Example 1: Uniform exponential convergence with bounded soft killing rate
• Example 2: Wasserstein distance
• Example 3: Counter-example to linear convergence when $h=0$ on some subset
• Example 4: Counter-example to uniform when no uniform convergence of semi-group
• Theorem 4
• proof