{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T07:25:26Z","timestamp":1774941926691,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,3,5]],"date-time":"2021-03-05T00:00:00Z","timestamp":1614902400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,3,5]],"date-time":"2021-03-05T00:00:00Z","timestamp":1614902400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["311945-2013"],"award-info":[{"award-number":["311945-2013"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2021,4]]},"DOI":"10.1007\/s00332-021-09686-5","type":"journal-article","created":{"date-parts":[[2021,3,5]],"date-time":"2021-03-05T20:02:21Z","timestamp":1614974541000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Periodic Solutions of Stochastic Differential Equations Driven by L\u00e9vy Noises"],"prefix":"10.1007","volume":"31","author":[{"given":"Xiao-Xia","family":"Guo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8482-3885","authenticated-orcid":false,"given":"Wei","family":"Sun","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,3,5]]},"reference":[{"key":"9686_CR1","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1016\/j.physleta.2015.09.046","volume":"380","author":"S Agarwal","year":"2016","unstructured":"Agarwal, S., Wettlaufer, J.S.: Maximal stochastic transport in the Lorenz equations. Phys. Lett. A 380, 142\u2013146 (2016)","journal-title":"Phys. Lett. A"},{"key":"9686_CR2","doi-asserted-by":"crossref","unstructured":"Applebaum, D.: L\u00e9vy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press (2009)","DOI":"10.1017\/CBO9780511809781"},{"key":"9686_CR3","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1016\/j.jmaa.2012.02.043","volume":"391","author":"J Bao","year":"2012","unstructured":"Bao, J., Yuan, C.: Stochastic population dynamics driven by L\u00e9vy noise. J. Math. Anal. Appl. 391, 363\u2013375 (2012)","journal-title":"J. Math. Anal. Appl."},{"key":"9686_CR4","doi-asserted-by":"publisher","first-page":"6601","DOI":"10.1016\/j.na.2011.06.043","volume":"74","author":"J Bao","year":"2011","unstructured":"Bao, J., Mao, X., Yin, G., Yuan, C.: Competitive Lotka\u2013Volterra population dynamics with jumps. Nonlinear Anal. 74, 6601\u20136616 (2011)","journal-title":"Nonlinear Anal."},{"key":"9686_CR5","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1016\/j.jde.2017.02.032","volume":"263","author":"F Chen","year":"2017","unstructured":"Chen, F., Han, Y., Li, Y., Yang, X.: Periodic solutions of Fokker\u2013Planck equations. J. Differ. Equ. 263, 285\u2013298 (2017)","journal-title":"J. Differ. Equ."},{"key":"9686_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/s11425-018-9527-1","author":"L Chen","year":"2019","unstructured":"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","journal-title":"Sci. China Math."},{"key":"9686_CR7","doi-asserted-by":"crossref","unstructured":"Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall\/CRC (2004)","DOI":"10.1201\/9780203485217"},{"key":"9686_CR8","doi-asserted-by":"crossref","unstructured":"Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press (1996)","DOI":"10.1017\/CBO9780511662829"},{"key":"9686_CR9","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1007\/s10959-008-0143-0","volume":"21","author":"Z Dong","year":"2008","unstructured":"Dong, Z.: On the uniqueness of invariant measure of the Burgers equation driven by L\u00e9vy processes. J. Theor. Probab. 21, 322\u2013335 (2008)","journal-title":"J. Theor. Probab."},{"key":"9686_CR10","first-page":"11","volume":"47","author":"Y Dong","year":"2018","unstructured":"Dong, Y.: Ergodicity of stochastic differential equations driven by L\u00e9vy noise with local Lipschitz coefficients. Adv. Math. 47, 11\u201347 (2018)","journal-title":"Adv. Math."},{"key":"9686_CR11","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1016\/j.jde.2011.03.015","volume":"251","author":"Z Dong","year":"2011","unstructured":"Dong, Z., Xie, Y.: Ergodicity of stochastic 2D Navier\u2013Stokes equation with L\u00e9vy noise. J. Differ. Equ. 251, 196\u2013222 (2011)","journal-title":"J. Differ. Equ."},{"key":"9686_CR12","first-page":"394","volume":"63","author":"JL Doob","year":"1948","unstructured":"Doob, J.L.: Asymptotic properties of Markoff transition probabilities. Trans. Am. Math. Soc. 63, 394\u2013421 (1948)","journal-title":"Trans. Am. Math. Soc."},{"key":"9686_CR13","unstructured":"Guo, X., Sun, W.: Periodic solutions of hybrid jump diffusion processes. arXiv:1911.07303 (2020)"},{"key":"9686_CR14","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/j.amc.2015.04.033","volume":"264","author":"G Hu","year":"2015","unstructured":"Hu, G., Li, Y.: Asymptotic behaviors of stochastic periodic differential equations with Markovian switching. Appl. Math. Comput. 264, 403\u2013416 (2015)","journal-title":"Appl. Math. Comput."},{"key":"9686_CR15","doi-asserted-by":"publisher","first-page":"896","DOI":"10.1016\/j.jmaa.2018.06.025","volume":"466","author":"H Hu","year":"2018","unstructured":"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\u2013926 (2018)","journal-title":"J. Math. Anal. Appl."},{"key":"9686_CR16","first-page":"66","volume":"4","author":"K It\u00f4","year":"1951","unstructured":"It\u00f4, K.: On stochastic differential equations. Mem. Am. Math. Soc. 4, 66 (1951)","journal-title":"Mem. Am. Math. Soc."},{"key":"9686_CR17","doi-asserted-by":"crossref","unstructured":"It\u00f4 K.: Differential equations determining a Markoff process (original Japanese: Zenkoku Sizyo Sugaku Danwakai) Stroock D.W., Varadhan S.R.S. (Eds.), Kiyoshi It\u00f4, Selected papers. Springer (1987)","DOI":"10.1007\/978-1-4612-5370-9_2"},{"key":"9686_CR18","doi-asserted-by":"crossref","unstructured":"Khasminskii, R.Z.: Stochastic Stability of Differential Equations, 2nd edn. Springer (2012)","DOI":"10.1007\/978-3-642-23280-0"},{"key":"9686_CR19","doi-asserted-by":"publisher","first-page":"1165","DOI":"10.4134\/JKMS.2013.50.6.1165","volume":"50","author":"D Li","year":"2013","unstructured":"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\u20131181 (2013)","journal-title":"J. Korean Math. Soc."},{"key":"9686_CR20","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","volume":"20","author":"EN Lorenz","year":"1963","unstructured":"Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130\u2013141 (1963)","journal-title":"J. Atmos. Sci."},{"key":"9686_CR21","unstructured":"\u00d8ksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions. Springer (2004)"},{"key":"9686_CR22","doi-asserted-by":"publisher","first-page":"1993","DOI":"10.1109\/TNN.2008.2005610","volume":"19","author":"A Patel","year":"2008","unstructured":"Patel, A., Kosko, B.: Stochastic resonance in continuous and spiking neutron models with Levy noise. IEEE Trans. Neural Netw. 19, 1993\u20132008 (2008)","journal-title":"IEEE Trans. Neural Netw."},{"key":"9686_CR23","doi-asserted-by":"crossref","unstructured":"Petersen, K.: Ergodic Theory. Cambridge University Press (1983)","DOI":"10.1017\/CBO9780511608728"},{"key":"9686_CR24","first-page":"375","volume":"3","author":"H Poincar\u00e9","year":"1881","unstructured":"Poincar\u00e9, H.: Memoire sur les courbes definier par une equation differentiate. J. Math. Pures Appl. 3, 375\u2013442 (1881)","journal-title":"J. Math. Pures Appl."},{"key":"9686_CR25","first-page":"251","volume":"3","author":"H Poincar\u00e9","year":"1882","unstructured":"Poincar\u00e9, H.: Memoire sur les courbes definier par une equation differentiate. J. Math. Pures Appl. 3, 251\u2013296 (1882)","journal-title":"J. Math. Pures Appl."},{"key":"9686_CR26","first-page":"167","volume":"4","author":"H Poincar\u00e9","year":"1885","unstructured":"Poincar\u00e9, H.: Memoire sur les courbes definier par une equation differentiate. J. Math. Pures Appl. 4, 167\u2013244 (1885)","journal-title":"J. Math. Pures Appl."},{"key":"9686_CR27","first-page":"151","volume":"4","author":"H Poincar\u00e9","year":"1886","unstructured":"Poincar\u00e9, H.: Memoire sur les courbes definier par une equation differentiate. J. Math. Pures Appl. 4, 151\u2013217 (1886)","journal-title":"J. Math. Pures Appl."},{"key":"9686_CR28","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1016\/j.bulsci.2009.01.003","volume":"134","author":"J Ren","year":"2010","unstructured":"Ren, J., Wu, J., Zhang, X.: Exponential ergodicity of multi-valued stochastic differential equations. Bull. Sci. Math. Fr. 134, 391\u2013404 (2010)","journal-title":"Bull. Sci. Math. Fr."},{"key":"9686_CR29","unstructured":"Situ, R.: Theory of Stochastic Differential Equations with Jumps and Applications. Springer (2006)"},{"key":"9686_CR30","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/s11118-011-9220-6","volume":"36","author":"B Xie","year":"2012","unstructured":"Xie, B.: Uniqueness of invariant measure of infinite dimensional stochastic differential equations driven by L\u00e9vy noise. Potent. Anal. 36, 35\u201366 (2012)","journal-title":"Potent. Anal."},{"key":"9686_CR31","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1214\/19-AIHP959","volume":"56","author":"L Xie","year":"2020","unstructured":"Xie, L., Zhang, X.: Ergodicity of stochastic differential equations with jumps and singular coefficients. Ann. Inst. H. Poincar\u00e9 Probab. Stat. 56, 175\u2013229 (2020)","journal-title":"Ann. Inst. H. Poincar\u00e9 Probab. Stat."},{"key":"9686_CR32","doi-asserted-by":"publisher","first-page":"1005","DOI":"10.3934\/dcds.2009.24.1005","volume":"24","author":"D Xu","year":"2009","unstructured":"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\u20131023 (2009)","journal-title":"Discrete Contin. Dyn. Syst. Ser. A"},{"key":"9686_CR33","doi-asserted-by":"publisher","first-page":"128","DOI":"10.1016\/j.na.2014.05.004","volume":"108","author":"D Xu","year":"2014","unstructured":"Xu, D., Li, B., Long, S., Teng, L.: Moment estimate and existence for solutions of stochastic functional differential equations. Nonlinear Anal. 108, 128\u2013143 (2014)","journal-title":"Nonlinear Anal."},{"key":"9686_CR34","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1016\/j.na.2014.10.039","volume":"114","author":"D Xu","year":"2015","unstructured":"Xu, D., Li, B., Long, S., Teng, L.: Corrigendum to \u201cMoment estimate and existence for solutions of stochastic functional differential equations\u2019\u2019. Nonlinear Anal. 114, 40\u201341 (2015)","journal-title":"Nonlinear Anal."},{"key":"9686_CR35","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/j.jmaa.2015.04.090","volume":"430","author":"X Zhang","year":"2015","unstructured":"Zhang, X., Wang, K., Li, D.: Stochastic periodic solutions of stochastic differential equations driven by L\u00e9vy process. J. Math. Anal. Appl. 430, 231\u2013242 (2015)","journal-title":"J. Math. Anal. Appl."},{"key":"9686_CR36","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1016\/j.nahs.2017.04.003","volume":"26","author":"S Zhang","year":"2017","unstructured":"Zhang, S., Meng, X., Feng, T., Zhang, T.: Dynamics analysis and numerical simulations of a stochastic non-autonomous predator\u2013prey system with impulsive effects. Nonlinear Anal. Hybrid Syst. 26, 19\u201337 (2017)","journal-title":"Nonlinear Anal. Hybrid Syst."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-021-09686-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-021-09686-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-021-09686-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,31]],"date-time":"2023-10-31T04:32:50Z","timestamp":1698726770000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-021-09686-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,5]]},"references-count":36,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["9686"],"URL":"https:\/\/doi.org\/10.1007\/s00332-021-09686-5","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,5]]},"assertion":[{"value":"23 June 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 February 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 March 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"32"}}