带有Stepanov概周期系数的无穷维随机微分方程的θ-概周期解

doi:10.6040/j.issn.1671-9352.0.2022.486

《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (6): 113-126.doi: 10.6040/j.issn.1671-9352.0.2022.486

• • 上一篇

带有Stepanov概周期系数的无穷维随机微分方程的θ-概周期解

陈叶君¹(),丁惠生^1,^2,*()

1. 江西师范大学数学与统计学院，江西南昌 330022
2. 江西省应用数学中心，江西南昌 330022

收稿日期:2022-09-09 出版日期:2023-06-20 发布日期:2023-05-23
通讯作者: 丁惠生 E-mail:chenyejun999@jxnu.edu.cn;dinghs@mail.ustc.edu.cn
作者简介:陈叶君(1995—)，男，博士研究生，研究方向为现代分析. E-mail：chenyejun999@jxnu.edu.cn
基金资助:
国家自然科学基金资助项目(11861037);江西省双千计划(jxsq2019201001);江西省教育厅研究生创新基金(YC2021-B078)

θ-almost periodic solutions for stochastic differential equations in infinite dimensions with Stepanov almost periodic coefficients

Yejun CHEN¹(),Huisheng DING^1,^2,*()

1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, Jiangxi, China
2. Jiangxi Provincial Center for Applied Mathematics, Nanchang 330022, Jiangxi, China

Received:2022-09-09 Online:2023-06-20 Published:2023-05-23
Contact: Huisheng DING E-mail:chenyejun999@jxnu.edu.cn;dinghs@mail.ustc.edu.cn

摘要/Abstract

摘要：

基于Raynaud de Fitte的最新工作，本文考虑了带有Stepanov概周期系数的无穷维随机微分方程 $\mathrm{d} X(t)=A X(t)\mathrm{d} t+F(t, X(t))\mathrm{d} t+G(t, X(t))\mathrm{d} W(t)$ 的概周期性。在更弱的条件下(A生成的C₀半群不必是压缩的，F、G是Stepanov概周期而不必是概周期的)，我们得到了该方程的θ-概周期解的存在性和唯一性，并且证明了该解是依路径分布概周期的。

关键词: 概周期, θ-概周期, Stepanov概周期, 依分布, 随机微分方程

Abstract:

Based on the work of Raynaud de Fitte, in this paper, we consider almost periodicity of stochastic differential equations in infinite dimensions with Stepanov almost periodic coefficients $\mathrm{d} X(t)=A X(t)\mathrm{d} t+F(t, X(t))\mathrm{d} t+G(t, X(t))\mathrm{d} W(t).$ Under the weaker conditions (C₀-semigroup generated by A is not necessarily contractive, and F, G can be Stepanov almost periodic and not necessarily almost periodic), we obtain the existence and uniqueness of θ-almost periodic solutions of these equations, and furthermore, we prove that the solution is almost periodic in path distribution.

Key words: almost periodic, θ-almost periodic, Stepanov almost periodic, in distribution, stochastic differential equations

中图分类号:

O211.63

陈叶君,丁惠生. 带有Stepanov概周期系数的无穷维随机微分方程的θ-概周期解[J]. 《山东大学学报(理学版)》, 2023, 58(6): 113-126.

Yejun CHEN,Huisheng DING. θ-almost periodic solutions for stochastic differential equations in infinite dimensions with Stepanov almost periodic coefficients[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(6): 113-126.

参考文献 29

1	BOHR H . Zur theorie der fast periodischen funktionen: Ⅰ[J]. Acta Mathematica, 1924, 45, 29- 127.
2	BOHR H . Zur Theorie der Fastperiodischen Funktionen: Ⅱ[J]. Acta Mathematica, 1925, 46, 101- 214. doi: 10.1007/BF02543859
3	BOHR H . Zur Theorie der fastperiodischen Funktionen: Ⅲ[J]. ActaMathematica, 1926, 47, 237- 281.
4	ARNOLD L , TUDOR C . Stationary and almost periodic solutions of almost periodic affine stochastic differential equations[J]. Stochastics and Stochastics Reports, 1998, 64 (3/4): 177- 193.
5	BEDOUHENE F , CHALLALI N , MELLAH O , et al. Almost automorphy and various extensions for stochastic processes[J]. Journal of Mathematical Analysis and Applications, 2015, 429 (2): 1113- 1152. doi: 10.1016/j.jmaa.2015.04.014
6	CHEBAN D , LIU Zhenxin . Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations[J]. Journal of Differential Equations, 2020, 269 (4): 3652- 3685. doi: 10.1016/j.jde.2020.03.014
7	CHEBAN D , LIU Zhenxin . Averaging principle on infinite intervals for stochastic ordinary differential equations[J]. Electronic Research Archive, 2021, 29 (4): 2791- 2817. doi: 10.3934/era.2021014
8	CHENG Mengyu , LIU Zhenxin . Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients[J]. Discrete and Continuous Dynamical Systems-B, 2021, 26 (12): 6425- 6462. doi: 10.3934/dcdsb.2021026
9	CUI Jing , SHEN Guangjun . Almost automorphic solutions to stochastic evolution equations driven by Lévy processes[J]. Chinese Journal of Applied Probability and Statistics, 2017, 33 (5): 450- 466. doi: 10.3969/j.issn.1001-4268.2017.05.002
10	DA PRATO G , TUDOR C . Periodic and almost periodic solutions for semilinear stochastic equations[J]. Stochastic Analysis and Applications, 1995, 13 (1): 13- 33. doi: 10.1080/07362999508809380
11	GU Yuanfang , REN Yong , SAKTHIVEL R . Square-mean pseudo almost automorphic mild solutions for stochastic evolution equations driven by G-Brownian motion[J]. Stochastic Analysis and Applications, 2016, 34 (3): 528- 545. doi: 10.1080/07362994.2016.1155159
12	KAMENSKⅡ M , MELLAH O , RAYNAUD DE FITTE P . Weak averaging of semilinear stochastic differential equations with almost periodic coefficients[J]. Journal of Mathematical Analysis and Applications, 2015, 427 (1): 336- 364. doi: 10.1016/j.jmaa.2015.02.036
13	LI Yong , LIU Zhenxin , WANG Wenhe . Almost periodic solutions and stable solution for stochastic differential equations[J]. Discrete and Continuous Dynamical Systems-B, 2019, 24 (11): 5927- 5944.
14	LIU Zhenxin , WANG Wenhe . Favard separation method for almost periodic stochastic differential equations[J]. Journal of Differential Equations, 2016, 260 (11): 8109- 8136. doi: 10.1016/j.jde.2016.02.019
15	MELLAH O , RAYNAUD DE FITTE P . Counter examples to mean square almost periodicity of the solutions of some SDEs with almost periodic coefficients[J]. Electronic Journal of Differential Equations, 2013, 91, 1- 7.
16	TUDOR C . Almost periodic solutions of affine stochastic evolution equations[J]. Stochastics and Stochastics Reports, 1992, 38 (4): 251- 266. doi: 10.1080/17442509208833758
17	RAYNAUD DE FITTE P . Almost periodicity and periodicity for nonautonomous random dynamical systems[J]. Stochastics and Dynamics, 2021, 21 (6): 1- 34.
18	STEPANOFF W . über einige Verallgemeinerungen der fast periodischen Funktionen[J]. Mathematische Annalen, 1926, 95 (1): 473- 498. doi: 10.1007/BF01206623
19	CHEN Chuanhua , LI Hongxu . Composition of S^p-weighted pseudo almost automorphic functions and applications[J]. Electronic Journal of Differential Equations, 2014, 236, 1- 15.
20	GILMORE M E , GUIVER C , LOGEMANN H . Incremental input-to-state stability for Lur'e systems and asymptotic behaviour in the presence of Stepanov almost periodic forcing[J]. Journal of Differential Equations, 2021, 300, 692- 733. doi: 10.1016/j.jde.2021.08.009
21	HE Bing , WANG Qiru . On completeness of the space of weighted Stepanov-like pseudo almost automorphic (periodic)functions[J]. Journal of Mathematical Analysis and Applications, 2018, 465 (2): 1176- 1185. doi: 10.1016/j.jmaa.2018.05.057
22	LEVITAN B M , ZHIKOV V V . Almost periodic functions and differential equations[M]. New York: Cambridge University Press, 1982.
23	AMERIO L , PROUSE G . Almost-periodic functions and functional equations[M]. New York: Van Nostrand Reinhold, 1971.
24	SELL G . Topological dynamics and ordinary differential equations[M]. New York: Van Nostrand Reinhold, 1971.
25	DUDLEY R M . Real analysis and probability[M]. Cambridge: Cambridge University Press, 2002.
26	BEDOUHENE F , MELLAH O , RAYNAUD DE FITTE P . Bochner-almost periodicity for stochastic processes[J]. Stochastic Analysis and Applications, 2012, 30 (2): 322- 342. doi: 10.1080/07362994.2012.649628
27	DUAN Jinqiao . An introduction to stochastic dynamics[M]. New York: Cambridge University Press, 2015.
28	DA PRATO G , ZABCZYK J . Stochastic equations in infinite dimensions[M].2nd ed New York: Cambridge University Press, 2014.
29	LIU Wei , RÖCKNER M . Stochastic partial differential equations: an introduction[M]. Berlin: Springer, 2015.

多维度评价

Viewed

Full text

Abstract

Cited

Shared

Discussed

[1]	王小焕,吕广迎,戴利杰. Gronwall不等式的推广及应用[J]. 《山东大学学报(理学版)》, 2022, 57(6): 94-101.
[2]	陈丽,林玲. 具有时滞效应的股票期权定价[J]. 山东大学学报（理学版）, 2018, 53(4): 36-41.
[3]	肖新玲. 由马氏链驱动的正倒向随机微分方程[J]. 山东大学学报（理学版）, 2018, 53(4): 46-54.
[4]	杨叙,李硕. 白噪声和泊松随机测度驱动的倒向重随机微分方程的比较定理[J]. 山东大学学报（理学版）, 2017, 52(4): 26-29.
[5]	聂天洋,史敬涛. 完全耦合正倒向随机控制系统的动态规划原理与最大值原理之间的联系[J]. 山东大学学报（理学版）, 2016, 51(5): 121-129.
[6]	王先飞, 江龙, 马娇娇. 具有Osgood型生成元的多维倒向重随机微分方程[J]. 山东大学学报（理学版）, 2015, 50(08): 24-33.
[7]	方瑞, 马娇娇, 范胜君. 一类倒向随机微分方程解的稳定性定理[J]. 山东大学学报（理学版）, 2015, 50(06): 39-44.
[8]	王春生, 李永明. 中立型多变时滞随机微分方程的稳定性[J]. 山东大学学报（理学版）, 2015, 50(05): 82-87.
[9]	郑石秋,冯立超,刘秋梅. 系数连续的反射倒向随机微分方程的#br# 表示定理与逆比较定理[J]. 山东大学学报（理学版）, 2014, 49(03): 107-110.
[10]	许晓明. 超前倒向随机微分方程的反射解及相应的最优停止问题[J]. J4, 2013, 48(6): 14-17.
[11]	孟祥波1,张立东1*,杜子平2. 随机利率下保险公司最优保费策略[J]. J4, 2013, 48(3): 106-110.
[12]	陈新一. 一类非自治捕食系统的动力学行为[J]. J4, 2013, 48(12): 18-23.
[13]	孙启良,张启侠*. 随机H2／H∞控制的最大值原理方法[J]. J4, 2013, 48(09): 90-95.
[14]	郭莹. 差分方程的伪概周期解[J]. J4, 2012, 47(2): 42-46.
[15]	石学军1,江龙2*. 连续生成元的一维反射倒向随机微分方程的L^p-解[J]. J4, 2012, 47(11): 119-126.

带有Stepanov概周期系数的无穷维随机微分方程的θ-概周期解

θ-almost periodic solutions for stochastic differential equations in infinite dimensions with Stepanov almost periodic coefficients

RichHTML

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献 29

相关文章 15

多维度评价

本文评价

推荐阅读 10