# American Institute of Mathematical Sciences

January  2014, 34(1): 79-98. doi: 10.3934/dcds.2014.34.79

## Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems

 1 School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023 2 Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China 3 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain 4 Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 07737 Jena

Received  November 2012 Revised  March 2013 Published  June 2013

This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a Hölder continuous function with Hölder exponent in $(1/2,1)$, and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion $B^H$ with Hurst parameter $H>1/2$. In contrast to the article by Maslowski and Nualart [17], we present here an existence and uniqueness result in the space of Hölder continuous functions with values in a Hilbert space $V$. If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the Hölder continuous functions. That space of functions is appropriate to introduce a non-autonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing $B^H$ as the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion.
Citation: Yong Chen, Hongjun Gao, María J. Garrido–Atienza, Björn Schmalfuss. Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 79-98. doi: 10.3934/dcds.2014.34.79
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##### References:
 [1] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473 [2] Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197 [3] Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257 [4] Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745 [5] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553 [6] Caibin Zeng, Xiaofang Lin, Jianhua Huang, Qigui Yang. Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise. Communications on Pure & Applied Analysis, 2020, 19 (2) : 811-834. doi: 10.3934/cpaa.2020038 [7] Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255 [8] Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199 [9] Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483 [10] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [11] Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084 [12] Monia Karouf. Reflected solutions of backward doubly SDEs driven by Brownian motion and Poisson random measure. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5571-5601. doi: 10.3934/dcds.2019245 [13] Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325 [14] Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 [15] Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077 [16] Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326 [17] Yousef Alnafisah, Hamdy M. Ahmed. Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021031 [18] Daoyi Xu, Weisong Zhou. Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2161-2180. doi: 10.3934/dcds.2017093 [19] Giuseppe Da Prato, Arnaud Debussche. Asymptotic behavior of stochastic PDEs with random coefficients. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1553-1570. doi: 10.3934/dcds.2010.27.1553 [20] Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122

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