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On the coming down from infinity of continuous-state branching processes with drift-interaction

Félix Rebotier

TL;DR

This work analyzes the boundary behavior of continuous-state branching processes with drift-interaction (CBDIs) in the presence of generalized drift and jumps. It develops Lyapunov-function and comparison-method techniques to obtain explicit, sufficient conditions for non-explosion and for coming down from infinity under relaxed moment assumptions on the Lévy measure. A key contribution is proving that, under a one-sided Lipschitz condition on the drift and non-explosion, the CBDI starting from infinity converges locally uniformly to the strong solution of a limiting SDE, and characterizing that SDE via a shifted Poisson framework. Additionally, the paper establishes regularity in the initial condition and provides a constructive description of the CBDI started at infinity, including a robust stochastic equation and uniqueness results with no negative jumps. Overall, it extends entrance-boundary analysis for CBDIs beyond finite first-moment Lévy measures and offers tools for studying boundary phenomena in interacting branching systems.

Abstract

We study the phenomenon of coming down from infinity - that is, when the process starts from infinity and never returns to it - for continuous-state branching processes with generalized drift. We provide sufficient conditions on the drift term and the branching mechanism to ensure both non-explosion and coming down from infinity, without requiring the associated jump measure to have a finite first moment. Assuming the process comes down from infinity and the drift satisfies a one-sided Lipschitz condition, we show that, as the initial values tend to infinity, the process converges locally uniformly almost surely to the strong solution of a stochastic differential equation. The main techniques employed are comparison principles for solutions of stochastic equations and the method of Lyapunov functions, the latter being briefly reviewed in a broader setting.

On the coming down from infinity of continuous-state branching processes with drift-interaction

TL;DR

This work analyzes the boundary behavior of continuous-state branching processes with drift-interaction (CBDIs) in the presence of generalized drift and jumps. It develops Lyapunov-function and comparison-method techniques to obtain explicit, sufficient conditions for non-explosion and for coming down from infinity under relaxed moment assumptions on the Lévy measure. A key contribution is proving that, under a one-sided Lipschitz condition on the drift and non-explosion, the CBDI starting from infinity converges locally uniformly to the strong solution of a limiting SDE, and characterizing that SDE via a shifted Poisson framework. Additionally, the paper establishes regularity in the initial condition and provides a constructive description of the CBDI started at infinity, including a robust stochastic equation and uniqueness results with no negative jumps. Overall, it extends entrance-boundary analysis for CBDIs beyond finite first-moment Lévy measures and offers tools for studying boundary phenomena in interacting branching systems.

Abstract

We study the phenomenon of coming down from infinity - that is, when the process starts from infinity and never returns to it - for continuous-state branching processes with generalized drift. We provide sufficient conditions on the drift term and the branching mechanism to ensure both non-explosion and coming down from infinity, without requiring the associated jump measure to have a finite first moment. Assuming the process comes down from infinity and the drift satisfies a one-sided Lipschitz condition, we show that, as the initial values tend to infinity, the process converges locally uniformly almost surely to the strong solution of a stochastic differential equation. The main techniques employed are comparison principles for solutions of stochastic equations and the method of Lyapunov functions, the latter being briefly reviewed in a broader setting.

Paper Structure

This paper contains 17 sections, 19 theorems, 160 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Condition [A]
  4. Main results
  5. Condition [B]
  6. First passage times and Lyapunov functions
  7. (i)
  8. (i)
  9. (ii)
  10. (iii)
  11. Proofs of the main results: coming down from infinity
  12. Lyapunov functions for the CBDI
  13. Regularity of the CBDI in the initial state
  14. The stochastic equation satisfied by $(X_t^{\infty}, \ t> 0)$
  15. Appendix
  16. ...and 2 more sections

Key Result

Proposition 2.1

Assume that $I$ satisfies [A]. Then, for all positive $\mathcal{F}_0$-measurable random variable $X_0$, there exists a unique càdlàg strong solution to eds, up to the random time $\zeta$.

Theorems & Definitions (34)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 3.1
  • Example 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Theorem 3.2
  • ...and 24 more