[Paper Review] Branching Random Walks with Immigration. Lyapunov Stability

Yu. Makarova, Dan Han+2 more|arXiv (Cornell University)|Feb 10, 2020

Stochastic processes and statistical mechanics3 references4 citations

TL;DR

This paper studies continuous-time branching random walks on a multidimensional lattice with immigration, focusing on Lyapunov stability under state-dependent intensities. It derives moment equations and asymptotic bounds for the first and second moments, showing that immigration stabilizes the process against extinction even when birth rates are subcritical, with second moments converging to a stable profile modulated by spatial intensity functions and diffusion coefficients.

ABSTRACT

We consider a continuous-time symmetric branching random walk on multidimensional lattices with immigration and infinite number of initial particles. We assume that at every lattice point a process of birth and death of particles is described by a Bienayme-Galton- Watson branching process. The assumption on immigration of particles is that a new particle can appear at every lattice point from the outside. The subject of the study is a limit distribution of the particle field on the lattice. The differential equations for correlation functions for the number of particles at an arbitrary time moment at the fixed points on the lattice are obtained. We study the asymptotic behaviour for the first two moments of a number of particles at every lattice point and Lyapunov stability of the process. The Lyapunov stability is considered under the assumptions that birth, death and immigration intensities depend on the lattice points.

Motivation & Objective

• To establish Lyapunov stability for branching random walks with immigration on a multidimensional lattice under time- and space-dependent intensities.
• To address the instability of standard critical branching processes under small perturbations by introducing immigration as a stabilizing mechanism.
• To derive and analyze the first and second moments of the particle field, accounting for spatial heterogeneity in birth, death, and immigration rates.
• To provide asymptotic estimates for the second moment that reflect long-time stability and convergence to a bounded profile despite perturbations.
• To demonstrate that the system reaches a statistical equilibrium under immigration, even when the underlying branching process would otherwise degenerate.

Proposed method

• Models the particle field $ n(t,x) $ as a continuous-time Markov process on $ \mathbb{Z}^d $, with particles undergoing symmetric random walks, binary or general branching, and immigration.
• Uses the generator $ \mathcal{L} $ of the underlying symmetric random walk with intensity $ \varkappa $, and assumes irreducibility and symmetric transition rates $ a(z) = a(-z) $.
• Derives a differential equation for the first moment $ m_1(t,x,y) $, showing exponential decay of initial conditions and convergence to a steady-state profile.
• Applies advanced stochastic calculus to derive a second-order differential equation for $ m_2(t,x,y) $, incorporating interactions between branching, diffusion, and immigration.
• Implements perturbation analysis with $ \varepsilon $-dependent bounds on intensity functions to derive upper and lower estimates for the second moment.
• Uses generating function techniques cautiously, noting limitations in capturing full moment dynamics under non-homogeneous intensities.

Experimental results

Research questions

• RQ1 How does immigration stabilize a subcritical or critical branching random walk on $ \mathbb{Z}^d $ under time- and space-dependent intensities?
• RQ2 What is the asymptotic behavior of the first and second moments of the particle field in the presence of immigration and spatial heterogeneity?
• RQ3 Can Lyapunov stability be established for the second moment when birth, death, and immigration rates vary across space?
• RQ4 How do perturbations in intensity functions affect the long-time behavior of the second moment and the system's stability?
• RQ5 What are the precise bounds on the second moment that reflect the interplay between diffusion, branching, and immigration in non-homogeneous environments?

Key findings

• The second moment $ m_2(t,x,y) $ is bounded above and below by expressions involving $ L(t,x,y) $, $ \frac{k_0^2}{v_0^2} $, and exponentially decaying terms in $ t $, with constants depending on $ \varepsilon $, $ k_0 $, $ \varkappa $, and $ v_0 $.
• The asymptotic behavior of the second moment converges to $ L(t,x,y) + \frac{k_0^2}{v_0^2} + O(\varepsilon) $ as $ t \to \infty $, indicating long-term stability.
• The upper and lower bounds for $ m_2(t,x,y) $ are expressed as $ m_2(t,x,y) \leq L(t,x,y) + \frac{k_0^2}{v_0^2} + \frac{k_0\varkappa}{v_0^2} + C_2^+ e^{-(v_0 - \varepsilon)t} + O(\varepsilon) $, with similar lower bounds.
• The term $ \frac{k_0^2}{v_0^2} $ represents the dominant contribution to the second moment in the steady state, arising from the immigration intensity $ k_0 $.
• Exponential decay terms $ e^{-(v_0 \pm \varepsilon)t} $ and $ e^{-2(v_0 \pm \varepsilon)t} $ dominate transient behavior, ensuring convergence to equilibrium.
• The system exhibits Lyapunov stability: small perturbations in intensity functions lead to bounded deviations in the second moment, with error terms controlled by $ O(\varepsilon) $.