Poisson Stable Solutions for Stochastic Differential Equations with Lévy Noise

• Published in: Acta mathematica Sinica. English series Vol. 38; no. 1; pp. 22 - 54
• Main Authors: Liu, Xin, Liu, Zhen Xin
• Format: Journal Article
• Language: English
• Published: Beijing Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01.01.2022; Springer Nature B.V
• Subjects: Differential equations; Mathematical analysis; Mathematics; Mathematics and Statistics; 34C27; 34C25; 60H10; quasi-periodic solution; 60G51; Lévy noise; asymptotic stability; Levitan almost periodic solution; periodic solution; Poisson stable solution; almost periodic solution; 37B20; almost automorphic solution; Stochastic differential equation; Birkhoff recurrent solution
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-021-0107-1

In this paper, we use a unified framework to study Poisson stable (including stationary, periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent and Poisson stable) solutions for semilinear stochastic differential equations driven by infinite dimensional Lévy noise with large jumps. Under suitable conditions on drift, diffusion and jump coefficients, we prove that there exist solutions which inherit the Poisson stability of coefficients. Further we show that these solutions are globally asymptotically stable in square-mean sense. Finally, we illustrate our theoretical results by several examples.