Distribution Dependent Birth-Death Processes: $\mathbb{W}_p$-Estimate, Ergodicity and Propagation of Chaos

TL;DR

The paper analyzes distribution-dependent birth-death processes on $\mathbb Z_+$ with generator ${\bf L}^{a,b}$, where birth/death rates depend on the current distribution. It proves well-posedness in $\mathscr P_1$, provides $\mathbb W_p$-estimates for $p\in(1,\infty)$, and establishes exponential ergodicity and uniform in time propagation of chaos for mean-field birth-death systems, using Lyapunov functions and coupling techniques. It also proves Lipschitz continuity of the push-forward map $P_t^*$ in $\mathscr P_p$ with explicit constants and derives quantitative chaos bounds for finite-N particle systems, culminating with an appendix that gives a general well-posedness criterion for inhomogeneous jump processes. The results extend DDSDE and mean-field particle system theories to more general distribution-dependent jump dynamics and set a foundation for future extensions to broader classes of distribution-dependent jump processes.

Abstract

For a class of time inhomogenous distribution dependent birth-death processes, we derive the well-posedness, $\mathbb{W}_p$-estimate, exponential ergodicity, and uniform in time propagation of chaos. These extend the corresponding results derived for distribution dependent SDEs and mean field particle systems. As preparation, a criterion on the well-posedness of inhomogenous jump process is presented in the end of the paper, which should be interesting by itself.

Theorems & Definitions (16)

• Definition 1.1
• Definition 1.2
• Example 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Theorem 4.1
• proof