无穷维随机微分方程的渐近概周期解

数学物理学报 ›› 2023, Vol. 43 ›› Issue (2): 341-354.

无穷维随机微分方程的渐近概周期解

陈叶君(),丁惠生^*()

江西师范大学数学与统计学院南昌 330022

收稿日期:2022-06-22 修回日期:2022-10-17 出版日期:2023-04-26 发布日期:2023-04-17
通讯作者: 丁惠生，E-mail: dinghs@mail.ustc.edu.cn
作者简介:陈叶君,E-mail: chenyejun999@jxnu.edu.cn
基金资助:
国家自然科学基金(11861037);江西省双千计划(jxsq2019201001);江西省教育厅研究生创新基金(YC2021-B078)

Asymptotically Almost Periodic Solutions for Stochastic Differential Equations in Infinite Dimensions

Chen Yejun(),Ding Huisheng()

School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022

Received:2022-06-22 Revised:2022-10-17 Online:2023-04-26 Published:2023-04-17
Supported by:
NSFC(11861037);Double Thousand Plan of Jiangxi Province(jxsq2019201001);Graduate Innovation Fund of Jiangxi Provincial Education Department(YC2021-B078)

摘要/Abstract

摘要：

该文引入了渐近 $\theta$ -概周期随机过程的概念, 并在算子半群理论框架下研究了一类带有渐近概周期系数的无穷维随机微分方程, 利用随机分析理论建立了此类随机微分方程渐近 $\theta$ -概周期解的存在性. 此外该文还引入了依路径分布渐近概周期过程的概念, 并证明了上述渐近 $\theta$ -概周期解还是依路径分布渐近概周期的. 值得注意的是, 在早期的研究结果中, 建立的均是更弱的一维分布渐近概周期解的存在性.

关键词: 依路径分布渐近概周期, 渐近 $\theta$ -概周期, 随机微分方程

Abstract:

In this paper, we introduce the notion of asymptotically $\theta$ -almost periodic stochastic process and study a class of stochastic differential matrixs in infinite dimensions with asymptotically almost periodic coefficients under the framework of operator semigroup theory. Using stochastic analysis theory, the existence of asymptotically $\theta$ -almost periodic solutions of these matrixs is established. In addition, the concept of asymptotically almost periodic process in path distribution is introduced, and we prove that the above solutions are also asymptotically almost periodic in path distribution. It is noteworthy that all the earlier related results only give the existence of asymptotically almost periodic solutions in one-dimensional distribution, which are weaker than asymptotically almost periodic solutions in path distribution.

Key words: Asymptotically almost periodic in path distribution, Asymptotically $\theta$ -almost periodic, Stochastic differential matrixs

中图分类号:

O211.63

陈叶君, 丁惠生. 无穷维随机微分方程的渐近概周期解[J]. 数学物理学报, 2023, 43(2): 341-354.

Chen Yejun, Ding Huisheng. Asymptotically Almost Periodic Solutions for Stochastic Differential Equations in Infinite Dimensions[J]. Acta mathematica scientia,Series A, 2023, 43(2): 341-354.

参考文献 30

[1]	Arnold L, Tudor C. Stationary and almost periodic solutions of almost periodic affine stochastic differential matrixs. Stochastics Stochastics Rep, 1998, 64(3/4): 177-193
[2]	Bedouhene F, Challali N, Mellah O, et al. Almost automorphy and various extensions for stochastic processes. J Math Anal Appl, 2015, 429(2): 1113-1152 doi: 10.1016/j.jmaa.2015.04.014
[3]	Bedouhene F, Mellah O, Raynaud de Fitte P. Bochner-almost periodicity for stochastic process. Stoc Anal Appl, 2012, 30(2): 322-342
[4]	Besicovitch A S. Almost Periodic Functions. New York: Dover Publications, 1955
[5]	Bochner S. A new approach to almost periodicity. Proc Nat Acad Sci USA, 1962, 48: 2039-2043 doi: 10.1073/pnas.48.12.2039
[6]	Bochner S. Beiträge zur Theorie der fastperiodischen Funktionen. (German) Math Ann, 1927, 96(1): 119-147
[7]	Bochner S, Von Neumann J. Almost periodic functions in groups. II. Trans Amer Math Soc, 1935, 37(1): 21-50
[8]	Che B D, Liu Z X. Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential matrixs. J Differential Equations, 2020, 269(4): 3652-3685 doi: 10.1016/j.jde.2020.03.014
[9]	Che B D, Liu Z X. Averaging principle on infinite intervals for stochastic ordinary differential matrixs. Electron Res Arch, 2021, 29(4): 2791-2817 doi: 10.3934/era.2021014
[10]	Cheng M Y, Liu Z X. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete Contin Dyn Syst Ser B, 2021, 26(12): 6425-6462 doi: 10.3934/dcdsb.2021026
[11]	陈叶君, 丁惠生, 简伟刚. 关于群上的概周期函数的几点注记. 江西师范大学学报(自然科学版), 2020, 44(2): 202-205
	Chen Y J, Ding H S, Jian W G. Some notes about almost periodic function on groups. Journal of Jiangxi Normal University (Natural Science), 2020, 44(2): 202-205
[12]	Da P G, Tudor C. Periodic and almost periodic solutions for semilinear stochastic matrixs. Stochastic Anal Appl, 1995, 13(1): 13-33 doi: 10.1080/07362999508809380
[13]	Da P G, Zabczyk J. Stochastic Equations in Infinite Dimensions. Second edition. Encyclopedia of Mathematics and its Applications, 152. Cambridge:Cambridge University Press, 2014
[14]	Duan J Q. An Introduction to Stochastic Dynamics. Cambridge Texts in Applied Mathematics. New York: Cambridge University Press, 2015
[15]	Dudley R M. Real Analysis and Probability. Cambridge: Cambridge University Press, 2002
[16]	Kamenskii M, Mellah O, Raynaud de Fitte P. Weak averaging of semilinear stochastic differential matrixs with almost periodic coefficients. J Math Anal Appl, 2015, 427(1): 336-364 doi: 10.1016/j.jmaa.2015.02.036
[17]	Levitan B M, Zhikov V V. Almost Periodic Functions and Differential Equations. Cambridge-New York: Cambridge University Press, 1982
[18]	Li Y, Liu Z X, Wang W H. Almost periodic solutions and stable solution for stochastic differential matrixs. Discrete Contin Dyn Syst Ser B, 2019, 24(11): 5927-5944
[19]	Liu W, Röckner M. Stochastic Partial Differential Equations:An Introduction. Cham: Springer, 2015
[20]	Liu Z X, Wang W H. Favard separation method for almost periodic stochastic differential matrixs. J Differential Equations, 2016, 260(11): 8109-8136 doi: 10.1016/j.jde.2016.02.019
[21]	Mellah O, Raynaud D F P. Counterexamples to mean square almost periodicity of the solutions of some SDEs with almost periodic coefficients. Electron J Differential Equations, 2013, (91): 1-7
[22]	Morozan T, Tudor C. Almost periodic solutions of affine Itô matrixs. Stochastic Anal Appl, 1989, 7(4): 451-474 doi: 10.1080/07362998908809194
[23]	Raynaud D F P. Almost periodicity and periodicity for nonautonomous random dynamical systems. Stoch Dyn, 2021, 21(6): 1-34
[24]	Sell G. Topological Dynamics and Ordinary Differential Equations. London: Van Nostrand Reinhold, 1971
[25]	Tudor C. Almost periodic solutions of affine stochastic evolution matrixs. Stochastics Stochastics Rep, 1992, 38(4): 251-266
[26]	Vârsan C. Stochastic differential matrixs and asymptotic almost periodic solutions. Rev Roumaine Math Pures Appl, 1990, 35(5): 485-493
[27]	Vârsan C. Asymptotic almost periodic solutions for stochastic differential matrixs//Zabczyk J. Stochastic Systems and Optimization. Berlin: Springer, 2005: 152-158
[28]	Von Neumann J. Almost periodic functions in a group. I Trans Amer Math Soc, 1934, 36(3): 445-492
[29]	Von Neumann J, Wigner E P. Minimally almost periodic groups. Ann of Math, 1940, 41: 746-750 doi: 10.2307/1968853
[30]	Zhang C Y. Almost Periodic Type Functions and Ergodicity. Beijing: Science Press Beijing; Dordrecht: Kluwer Academic Publishers, 2003

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