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Periodic Solutions of Stochastic Differential Equations Driven by Lévy Noises

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Abstract

In this paper, we first show the well-posedness of SDEs driven by Lévy noises under mild conditions. Then, we consider the existence and uniqueness of periodic solutions of the SDEs. To establish the ergodicity and uniqueness of periodic solutions, we investigate the strong Feller property and the irreducibility of the corresponding time-inhomogeneous semigroups when both small and large jumps are allowed in the equations. Doob’s celebrated theorem on the uniqueness of invariant measures for time-homogeneous Markov processes has been generalized to obtain the uniqueness of periodic measures for time-inhomogeneous Markov processes. Some examples are presented to illustrate our results.

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References

  • Agarwal, S., Wettlaufer, J.S.: Maximal stochastic transport in the Lorenz equations. Phys. Lett. A 380, 142–146 (2016)

    Article  MathSciNet  Google Scholar 

  • Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press (2009)

  • Bao, J., Yuan, C.: Stochastic population dynamics driven by Lévy noise. J. Math. Anal. Appl. 391, 363–375 (2012)

    Article  MathSciNet  Google Scholar 

  • Bao, J., Mao, X., Yin, G., Yuan, C.: Competitive Lotka–Volterra population dynamics with jumps. Nonlinear Anal. 74, 6601–6616 (2011)

    Article  MathSciNet  Google Scholar 

  • Chen, F., Han, Y., Li, Y., Yang, X.: Periodic solutions of Fokker–Planck equations. J. Differ. Equ. 263, 285–298 (2017)

    Article  MathSciNet  Google Scholar 

  • Chen, L., Dong, Z., Jiang, J., Zhai, J.: On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity. Sci. China Math. (2019). https://doi.org/10.1007/s11425-018-9527-1

    Article  MATH  Google Scholar 

  • Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall/CRC (2004)

  • Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press (1996)

  • Dong, Z.: On the uniqueness of invariant measure of the Burgers equation driven by Lévy processes. J. Theor. Probab. 21, 322–335 (2008)

    Article  Google Scholar 

  • Dong, Y.: Ergodicity of stochastic differential equations driven by Lévy noise with local Lipschitz coefficients. Adv. Math. 47, 11–47 (2018)

    MathSciNet  MATH  Google Scholar 

  • Dong, Z., Xie, Y.: Ergodicity of stochastic 2D Navier–Stokes equation with Lévy noise. J. Differ. Equ. 251, 196–222 (2011)

    Article  Google Scholar 

  • Doob, J.L.: Asymptotic properties of Markoff transition probabilities. Trans. Am. Math. Soc. 63, 394–421 (1948)

    MATH  Google Scholar 

  • Guo, X., Sun, W.: Periodic solutions of hybrid jump diffusion processes. arXiv:1911.07303 (2020)

  • Hu, G., Li, Y.: Asymptotic behaviors of stochastic periodic differential equations with Markovian switching. Appl. Math. Comput. 264, 403–416 (2015)

    MathSciNet  MATH  Google Scholar 

  • Hu, H., Xu, L.: Existence and uniqueness theorems for periodic Markov process and applications to stochastic functional differential equations. J. Math. Anal. Appl. 466, 896–926 (2018)

    Article  MathSciNet  Google Scholar 

  • Itô, K.: On stochastic differential equations. Mem. Am. Math. Soc. 4, 66 (1951)

    Google Scholar 

  • Itô K.: Differential equations determining a Markoff process (original Japanese: Zenkoku Sizyo Sugaku Danwakai) Stroock D.W., Varadhan S.R.S. (Eds.), Kiyoshi Itô, Selected papers. Springer (1987)

  • Khasminskii, R.Z.: Stochastic Stability of Differential Equations, 2nd edn. Springer (2012)

  • Li, D., Xu, D.: Periodic solutions of stochastic delay differential equations and applications to Logistic equation and neural networks. J. Korean Math. Soc. 50, 1165–1181 (2013)

    Article  MathSciNet  Google Scholar 

  • Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  MathSciNet  Google Scholar 

  • Øksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions. Springer (2004)

  • Patel, A., Kosko, B.: Stochastic resonance in continuous and spiking neutron models with Levy noise. IEEE Trans. Neural Netw. 19, 1993–2008 (2008)

    Article  Google Scholar 

  • Petersen, K.: Ergodic Theory. Cambridge University Press (1983)

  • Poincaré, H.: Memoire sur les courbes definier par une equation differentiate. J. Math. Pures Appl. 3, 375–442 (1881)

    MATH  Google Scholar 

  • Poincaré, H.: Memoire sur les courbes definier par une equation differentiate. J. Math. Pures Appl. 3, 251–296 (1882)

    MATH  Google Scholar 

  • Poincaré, H.: Memoire sur les courbes definier par une equation differentiate. J. Math. Pures Appl. 4, 167–244 (1885)

    Google Scholar 

  • Poincaré, H.: Memoire sur les courbes definier par une equation differentiate. J. Math. Pures Appl. 4, 151–217 (1886)

    Google Scholar 

  • Ren, J., Wu, J., Zhang, X.: Exponential ergodicity of multi-valued stochastic differential equations. Bull. Sci. Math. Fr. 134, 391–404 (2010)

    Article  Google Scholar 

  • Situ, R.: Theory of Stochastic Differential Equations with Jumps and Applications. Springer (2006)

  • Xie, B.: Uniqueness of invariant measure of infinite dimensional stochastic differential equations driven by Lévy noise. Potent. Anal. 36, 35–66 (2012)

    Article  Google Scholar 

  • Xie, L., Zhang, X.: Ergodicity of stochastic differential equations with jumps and singular coefficients. Ann. Inst. H. Poincaré Probab. Stat. 56, 175–229 (2020)

    Article  MathSciNet  Google Scholar 

  • Xu, D., Huang, Y., Yang, Z.: Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete Contin. Dyn. Syst. Ser. A 24, 1005–1023 (2009)

    Article  MathSciNet  Google Scholar 

  • Xu, D., Li, B., Long, S., Teng, L.: Moment estimate and existence for solutions of stochastic functional differential equations. Nonlinear Anal. 108, 128–143 (2014)

    Article  MathSciNet  Google Scholar 

  • Xu, D., Li, B., Long, S., Teng, L.: Corrigendum to “Moment estimate and existence for solutions of stochastic functional differential equations’’. Nonlinear Anal. 114, 40–41 (2015)

    Article  MathSciNet  Google Scholar 

  • Zhang, X., Wang, K., Li, D.: Stochastic periodic solutions of stochastic differential equations driven by Lévy process. J. Math. Anal. Appl. 430, 231–242 (2015)

    Article  MathSciNet  Google Scholar 

  • Zhang, S., Meng, X., Feng, T., Zhang, T.: Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects. Nonlinear Anal. Hybrid Syst. 26, 19–37 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the Graduate Joint Training Program of the Guangdong Educational Department, China, and the Natural Sciences and Engineering Research Council of Canada (Grant No. 311945-2013). We thank the editor, the associate editor and the two referees for the careful reading of our paper and all of the insightful suggestions and comments that greatly improved the presentation of the paper.

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, China

    Xiao-Xia Guo

  2. Department of Mathematics and Statistics, Concordia University, Montreal, H3G 1M8, Canada

    Wei Sun

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  1. Xiao-Xia Guo
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  2. Wei Sun
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Correspondence to Wei Sun.

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Communicated by Amy Radunskaya.

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Guo, XX., Sun, W. Periodic Solutions of Stochastic Differential Equations Driven by Lévy Noises. J Nonlinear Sci 31, 32 (2021). https://doi.org/10.1007/s00332-021-09686-5

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