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A localized coupling approach to interacting continuous-state branching processes

Shukai Chen, Pei-Sen Li, Jian Wang

Abstract

We introduce a class of continuous-state branching processes with immigration, predation and competition, which can be viewed as a combination of the classical Lotka-Volterra model and continuous-state branching processes with competition that were introduced by Berestycki, Fittipaldi, and Fontbona (Probab. Theory Relat. Fields, 2018). This model can be constructed as a unique strong solution to a class of two-dimensional stochastic differential equations with jumps. We establish sharp conditions for the uniform ergodicity in the total variation of this model. Our proof relies on a novel, localized Markovian coupling approach, which is of its own interest in the ergodicity theory of Markov processes with interactions.

A localized coupling approach to interacting continuous-state branching processes

Abstract

We introduce a class of continuous-state branching processes with immigration, predation and competition, which can be viewed as a combination of the classical Lotka-Volterra model and continuous-state branching processes with competition that were introduced by Berestycki, Fittipaldi, and Fontbona (Probab. Theory Relat. Fields, 2018). This model can be constructed as a unique strong solution to a class of two-dimensional stochastic differential equations with jumps. We establish sharp conditions for the uniform ergodicity in the total variation of this model. Our proof relies on a novel, localized Markovian coupling approach, which is of its own interest in the ergodicity theory of Markov processes with interactions.

Paper Structure

This paper contains 14 sections, 19 theorems, 172 equations.

Table of Contents

  1. Introduction and main results
  2. Background and motivation
  3. Main result
  4. Approach and novelties
  5. Markovian coupling for the CBIPC-process
  6. Proof of Theorem \ref{['main result']}
  7. Proof of Proposition \ref{['prop:1.6']}
  8. Proof of Proposition \ref{['prop:1.7']}
  9. Local comparison principles
  10. Tail estimates of stopping times $\bar{\zeta}_0$ and $\tau^+_{2M}$
  11. Proof of Proposition $\ref{['prop:1.7']}$
  12. Proof of Proposition \ref{['prop:1.8']}
  13. Proof of Remark \ref{['remark on k le 0']}
  14. Appendix

Key Result

Proposition 1.1

There exists a unique nonnegative strong solution $(X_t,Y_t)_{t\ge0}$ to (main sde). $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (22)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Remark 1.9
  • Proposition 2.1
  • ...and 12 more