Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundary behavior at infinity for simple exchangeable fragmentation-coagulation process in critical slow regime

Lina Ji, Xiaowen Zhou

TL;DR

The paper analyzes the boundary behavior at infinity for a critical slow-regime simple EFC process, where coagulation and fragmentation occur at comparable rates. By linking the block-counting process to its generator and applying refined boundary-criteria through carefully chosen test functions, it establishes phase transitions in the infinity-boundary type that depend on the asymptotics of the rate-difference \\Phi_\\Lambda(n) - \\Phi_\\mu(n), including precise conditions for entrance vs exit boundaries, coming down from infinity, and explosion. The authors further develop a coupling- and localization-based framework to derive sharp results under different regimes of the tail parameter \alpha and coagulation smoothness, contributing a rigorous boundary-classification methodology for slow EFC dynamics. These results advance understanding of long-time and infinite-particle behavior in exchangeable fragmentation-coagulation systems with potential applications to related coalescent- fragmentation models.

Abstract

For a critical simple exchangeable fragmentation-coagulation process in slow regime where the coagulation rate and fragmentation rate are of the same order, we show that there exist phase transitions for its boundary behavior at infinity depending on the asymptotics of the difference between the two rates, and find rather sharp conditions for different boundary behaviors.

Boundary behavior at infinity for simple exchangeable fragmentation-coagulation process in critical slow regime

TL;DR

The paper analyzes the boundary behavior at infinity for a critical slow-regime simple EFC process, where coagulation and fragmentation occur at comparable rates. By linking the block-counting process to its generator and applying refined boundary-criteria through carefully chosen test functions, it establishes phase transitions in the infinity-boundary type that depend on the asymptotics of the rate-difference \\Phi_\\Lambda(n) - \\Phi_\\mu(n), including precise conditions for entrance vs exit boundaries, coming down from infinity, and explosion. The authors further develop a coupling- and localization-based framework to derive sharp results under different regimes of the tail parameter \alpha and coagulation smoothness, contributing a rigorous boundary-classification methodology for slow EFC dynamics. These results advance understanding of long-time and infinite-particle behavior in exchangeable fragmentation-coagulation systems with potential applications to related coalescent- fragmentation models.

Abstract

For a critical simple exchangeable fragmentation-coagulation process in slow regime where the coagulation rate and fragmentation rate are of the same order, we show that there exist phase transitions for its boundary behavior at infinity depending on the asymptotics of the difference between the two rates, and find rather sharp conditions for different boundary behaviors.

Paper Structure

This paper contains 11 sections, 32 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Simple EFC processes and main results
  3. Introduction of simple EFC processes
  4. Main Results
  5. Preliminaries
  6. Some known results
  7. Estimations of $\Lambda$ and $\mu$
  8. Some criteria for boundary classification
  9. Criteria for Markov processes on $\mathbb{N}_+$
  10. Criteria for block counting processes on $\bar{\mathbb{N}}_+$
  11. Proofs of the main results

Key Result

Theorem 1.4

Assume that $\mu(n) \underset{n \rightarrow \infty}{\sim} b(\log n)^\alpha n^{-2}$ and $\Phi_\Lambda(n) \underset{n \rightarrow\infty}{\sim} dn(\log n)^\beta$ for $b, d, \alpha> 0$ and $\beta > 1$.

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Theorem 3.9 of FZ22
  • Proposition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.7
  • Corollary 2.8
  • ...and 26 more