无穷维随机微分方程的渐近概周期解

Abstract

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

CLC Number:

O211.63

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.

Trendmd

References

[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
[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
[6]	Bochner S. Beitr?ge 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
[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
[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
[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
[11]	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
[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
[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, R?ckner 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
[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 It? matrixs. Stochastic Anal Appl, 1989, 7(4): 451-474
[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]	Varsan C. Stochastic differential matrixs and asymptotic almost periodic solutions. Rev Roumaine Math Pures Appl, 1990, 35(5): 485-493
[27]	Varsan 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
[30]	Zhang C Y. Almost Periodic Type Functions and Ergodicity. Beijing: Science Press Beijing; Dordrecht: Kluwer Academic Publishers, 2003

Asymptotically Almost Periodic Solutions for Stochastic Differential Equations in Infinite Dimensions

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 13

Metrics

Comments

Recommended 0

[1]	Wang Qiufen, Zhang Shuwen. Stochastic Predator-Prey System with Fear Effect and Holling-Type III Functional Response [J]. Acta mathematica scientia,Series A, 2024, 44(5): 1380-1391.
[2]	Yang Hong, Zhang Xiaoguang. A Stochastic SIS Epidemic Model on Simplex Complexes [J]. Acta mathematica scientia,Series A, 2024, 44(5): 1392-1399.
[3]	Junfeng Liu,Lei Mao,Zhi Wang. Moment Bounds for the Fractional Stochastic Heat Equation with Spatially Inhomogeneous White Noise [J]. Acta mathematica scientia,Series A, 2022, 42(4): 1186-1208.
[4]	Liya Liu,Daqing Jiang. Global Dynamics of a Stochastic Chemostat Model with General Response Function and Wall Growth [J]. Acta mathematica scientia,Series A, 2021, 41(6): 1912-1924.
[5]	Zhonghua Zhang,Qian Zhang. Qualitative Analysis of a Stochastic SIVS Epidemic Model with Nonlinear Perturbations Under Regime Switching [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1218-1234.
[6]	Linjuan Pu,Xiaozhong Yang,Shuzhen Sun. Numerical Analysis of a Class of Fractional Langevin Equation by Predictor-Corrector Method [J]. Acta mathematica scientia,Series A, 2020, 40(4): 1018-1028.
[7]	Qikang Ran. SDE Driven by Fractional Brown Motion and Their Coefficients are Locally Linear Growth [J]. Acta mathematica scientia,Series A, 2020, 40(1): 200-211.
[8]	Yingjie Fu,Guijie Lan,Shuwen Zhang,Chunjin Wei. Dynamics of a Stochastic Predator-Prey Model with Pulse Input in a Polluted Environment [J]. Acta mathematica scientia,Series A, 2019, 39(3): 674-688.
[9]	Guijie Lan,Yingjie Fu,Chunjin Wei,Shuwen Zhang. Stationary Distribution and Periodic Solution for Stochastic Predator-Prey Systems with Holling-Type Ⅲ Functional Response [J]. Acta mathematica scientia,Series A, 2018, 38(5): 984-1000.
[10]	Wei Fengying, Lin Qingteng. Extinction and Distribution for an SIQS Epidemic Model with Quarantined-Adjusted Incidence [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1148-1161.
[11]	Liangquan Zhang. The Viability Property for Path-Dependent SDE Under Open Constraints [J]. Acta mathematica scientia,Series A, 2015, 35(6): 1168-1179.
[12]	Hao Tao. A Conditional Variance for g-Expectation [J]. Acta mathematica scientia,Series A, 2015, 35(5): 995-1003.
[13]	Diem Dang Huan, Gao Hongjun. Well-Posedness for Fractional Neutral Impulsive Stochastic Integro-Differential Equations [J]. Acta mathematica scientia,Series A, 2015, 35(2): 405-421.