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

TL;DR

This paper studies a class of time-inhomogeneous distribution-dependent birth-death processes, establishing their well-posedness, stability, ergodic behavior, and propagation of chaos, extending known results from related stochastic systems.

Contribution

It introduces new results on well-posedness, $ ext{W}_p$-estimates, ergodicity, and propagation of chaos for inhomogeneous distribution-dependent birth-death processes, broadening the scope of existing theories.

Findings

Established well-posedness criteria for inhomogeneous jump processes

Proved exponential ergodicity of the processes

Demonstrated uniform in time propagation of chaos

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.