Extinction, explosion and contraction for time-inhomogeneous SDEs with jumps

TL;DR

This work analyzes time-inhomogeneous SDEs with jumps and nonnegative solutions, establishing existence and pathwise uniqueness for the strong solution of $X_t$ and deriving sharp thresholds for extinction, explosion, and contractivity. The authors develop a coupling framework combining reflection for diffusion and basic coupling for jumps to obtain explicit exponential contraction in a weighted-total variation distance, under Lyapunov-type drift and growth conditions. The results extend known criteria from time-homogeneous settings to time-inhomogeneous environments and nonlinear jump structures, with applications to killed mean-field SDEs. Methodologically, the paper leverages Itô calculus, Yamada–Watanabe arguments, and a carefully crafted coupling operator to yield quantitative ergodic rates and rigorous long-time behavior insights. These findings have implications for population dynamics models and interacting particle systems under varying environments.

Abstract

For a class of time-inhomogeneous SDEs with jumps, we establish criteria for the existence and uniqueness of the nonnegative solutions, and examine the extinction, the explosion together with the contractivity of the solutions, which generalize and improve upon earlier results in the literature. As an application, we study the aforementioned properties for a class of mean field SDEs.

Theorems & Definitions (22)

• Definition 2.1
• Definition 2.2
• Theorem 2.4
• Lemma 2.5
• Proposition 2.6
• Proposition 2.7
• Theorem 3.1
• Theorem 3.2
• Theorem 3.2
• Theorem 3.3