Stationary Measures for Stochastic Differential Equations with Jumps

TL;DR

This paper investigates stationary measures for stochastic differential equations with jumps, establishing existence under broad conditions and providing explicit solutions in special cases, enhancing understanding of long-term behavior.

Contribution

It introduces general conditions for the existence of stationary measures and explicitly characterizes them in specific cases using Fokker-Planck equations.

Findings

Existence of stationary measures proved under broad conditions.

Stationary measures characterized by Fokker-Planck equations in special cases.

Long-term distribution limits of system states identified.

Abstract

In the paper, stationary measures of stochastic differential equations with jumps are considered. Under some general conditions, existence of stationary measures is proved through Markov measures and Lyapunov functions. Moreover, for two special cases, stationary measures are given by solutions of Fokker-Planck equations and long time limits for the distributions of system states.