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Exponential Ergodicity of CBIRE-Processes with Competition and Catastrophes

Shukai Chen, Rongjuan Fang, Lina Ji, Jian Wang

Abstract

We establish the exponential ergodic property in a weighted total variation distance of continuous-state branching processes with immigration in random environments with competition and catastrophes, under a Lyapunov-type condition and other mild assumptions. The proof is based on a Markov coupling process along with some delicate estimates for the associated coupling generator. In particular, the main result indicates whether and how the competition mechanism, the environment and the catastrophe could balance the branching mechanism respectively to guarantee the exponential ergodicity of the process.

Exponential Ergodicity of CBIRE-Processes with Competition and Catastrophes

Abstract

We establish the exponential ergodic property in a weighted total variation distance of continuous-state branching processes with immigration in random environments with competition and catastrophes, under a Lyapunov-type condition and other mild assumptions. The proof is based on a Markov coupling process along with some delicate estimates for the associated coupling generator. In particular, the main result indicates whether and how the competition mechanism, the environment and the catastrophe could balance the branching mechanism respectively to guarantee the exponential ergodicity of the process.
Paper Structure (12 sections, 11 theorems, 131 equations)

This paper contains 12 sections, 11 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Unique strong solution and its coupling process
  5. Existence and uniqueness of strong solution
  6. Markov coupling process
  7. General result for the exponential ergodicity
  8. Estimate of the coupling generator
  9. Preliminary estimation of the coupling generator
  10. Detailed estimation of the coupling generator
  11. Further estimation of the coupling generator based on $S_1$
  12. Proofs of Theorem \ref{['main result']} and Theorem \ref{['example']}

Key Result

Theorem 1.1

Suppose that in the SDE (X) the rate function $r(x)$ is globally Lipschitz and $\alpha>0$. Let $(X_t)_{t\geq0}$ be a unique strong solution to (X). Let $V(x)=(x+1)^{\theta}$ with $\theta\in (0,1)$. Assume that where Suppose that one of the following assumptions holds: Then the process $(X_t)_{t\geq0}$ is exponentially ergodic in terms of the distance $W_V$, if

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Theorem 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Proposition 3.8
  • ...and 1 more