Continuous-state branching processes with L\'evy-Khintchine drift-interaction: Laplace duality and Fellerian extensions

TL;DR

This paper studies continuous-state branching processes with Le9vy-Khintchine drift, revealing duality properties, boundary behaviors, and extending the processes to include boundary re-entries, with applications to various boundary regimes.

Contribution

It introduces a Laplace duality for CBDI processes, constructs Fellerian extensions at boundaries, and characterizes boundary behaviors using regular variation assumptions.

Findings

Laplace duality relates CBDI processes with exchanged mechanisms.

Fellerian extension allows boundary re-entries under non-Lipschitz drift.

Parameters determine boundary regimes like entrance, exit, or regular.

Abstract

We investigate the class of continuous-state branching processes with interaction driven by a L\'evy-Khintchine type drift (CBDI). These $[0,\infty]$-valued processes capture both dynamics of branching and density-dependence, allowing for cooperation at low population sizes and competition at high densities. Although the interaction breaks the branching property, the L\'evy--Khintchine form of the drift induces a Laplace duality. This duality expresses the Laplace transform of a CBDI process in terms of that of another CBDI process, in which the branching and drift-interaction mechanisms are exchanged. The process, stopped upon hitting either boundary $0$ or $\infty$, is uniquely characterized in law by these mechanisms. A Fellerian extension is constructed when the drift is non-Lipschitz and sufficiently strong at a boundary, allowing the process to leave this boundary continuously and possibly re-enter it. We identify parameters, defined in terms of the mechanisms and their associated scale function and potential measure, that determine the boundary behavior at $0$ and $\infty$ (entrance, exit or regular). Settings exhibiting all regimes, including regular-for-itself and non-sticky boundaries, arise when the mechanisms are assumed to be regularly varying. Our approach combines Laplace duality and comparison principles. The duality facilitates the analysis of semigroups and the construction of sharp Lyapunov functions. Comparisons ensure monotonicity and convergence properties of first-passage times.