# American Institute of Mathematical Sciences

March  2020, 25(3): 1141-1158. doi: 10.3934/dcdsb.2019213

## Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion

 1 School of Mathematical Sciences, Fudan University, Shanghai, 200433, China 2 Graduate School of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan 3 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710072, China 4 School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

* Corresponding author: Yong Xu

Received  November 2018 Revised  February 2019 Published  March 2020 Early access  September 2019

Fund Project: The research of B. Pei was partially supported by the NSF of China (11802216), the China Postdoctoral Science Foundation (2019M651334) and JSPS KAKENHI Grant Number JP18F18314 (Grant-in-Aid for JSPS Fellows). The research of Y. Xu was partially supported by the NSF of China (11702216), Shaanxi Province Project for Distinguished Young Scholars and the Fundamental Research Funds for the Central Universities. B. Pei would like to thank JSPS for Postdoctoral Fellowships for Research in Japan (Standard).

In this paper, we focus on fast-slow stochastic partial differential equations in which the slow variable is driven by a fractional Brownian motion and the fast variable is driven by an additive Brownian motion. We establish an averaging principle in which the fast-varying diffusion process will be averaged out with respect to its stationary measure in the limit process. It is shown that the slow-varying process $L^{p}(p \geq 2)$ converges to the solution of the corresponding averaging equation. To reduce the complexity, one can concentrate on the limit process instead of studying the original full fast-slow system.

Citation: 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 and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
##### References:
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Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047. [7] S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction–diffusion equations, Probability Theory and Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z. [8] P. Chow, Stochastic Partial Differential Equations, Second edition. Advances in Applied Mathematics. CRC Press, Boca Raton, FL, 2015. [9] I. Chueshov and B. Schmalfuss, Averaging of attractors and inertial manifolds for parabolic PDE with random coefficients, Advanced Nonlinear Studies, 5 (2005), 461-492.  doi: 10.1515/ans-2005-0402. [10] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513. [11] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, Amsterdam, 2014. [12] T. Duncan, Y. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion i. theory, SIAM Journal on Control and Optimization, 38 (2000), 582-612.  doi: 10.1137/S036301299834171X. [13] M. Freidlin and A. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4684-0176-9. [14] H. Fu and J. Liu, Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations, Journal of Mathematical Analysis and Applications, 384 (2011), 70-86.  doi: 10.1016/j.jmaa.2011.02.076. [15] H. Fu, L. Wan and J. Liu, Strong convergence in averaging principle for stochastic hyperbolic–parabolic equations with two time-scales, Stochastic Processes and their Applications, 125 (2015), 3255-3279.  doi: 10.1016/j.spa.2015.03.004. [16] D. 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Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, Journal of Functional Analysis, 202 (2003), 277-305.  doi: 10.1016/S0022-1236(02)00065-4. [22] J. Mémin, Y. Mishura and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statistics & Probability Letters, 51 (2001), 197-206.  doi: 10.1016/S0167-7152(00)00157-7. [23] Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, Berlin, 2008. doi: 10.1007/978-3-540-75873-0. [24] D. Nualart, The Malliavin Calculus and Related Topics, second edition, Springer-Verlag, Berlin, 2006. [25] B. Øksendal, Stochastic Differential Equations, Springer, Heidelberg, 2003. [26] B. Pei, Y. Xu and J. Wu, Two-time-scales hyperbolic-parabolic equations driven by Poisson random measures: Existence, uniqueness and averaging principles,, Journal of Mathematical Analysis and Applications, 447 (2017), 243-268.  doi: 10.1016/j.jmaa.2016.10.010. [27] B. Pei, Y. Xu and G. Yin, Averaging principles for SPDEs driven by fractional Brownian motions with random delays modulated by two-time-scale Markov switching processes, Stochastics & Dynamics, 18 (2018), 1850023, 19pp. doi: 10.1142/S0219493718500235. [28] B. Pei, Y. Xu and G. Yin, Stochastic averaging for a class of two-time-scale systems of stochastic partial differential equations, Nonlinear Analysis: Theory Method and Application, 160 (2017), 159-176.  doi: 10.1016/j.na.2017.05.005. [29] B. Pei, Y. Xu, G. Yin and X. Zhang, Averaging principles for functional stochastic partial differential equations driven by a fractional Brownian motion modulated by two-time-scale Markovian switching processes, Nonlinear Analysis: Hybrid Systems, 27 (2018), 107-124.  doi: 10.1016/j.nahs.2017.08.008. [30] J. Seidler, Da Prato-Zabczyk's maximal inequality revisited I, Mathematica Bohemica, 118 (1993), 67-106. [31] W. Thompson, R. Kuske and A. Monahan, Stochastic averaging of dynamical systems with multiple time scales forced with α-stable noise, Multiscale Modeling & Simulation, 13 (2015), 1194-1223.  doi: 10.1137/140990632. [32] S. Tindel, C. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probability Theory and Related Fields, 127 (2003), 186-204.  doi: 10.1007/s00440-003-0282-2. [33] J. Xu, Y. Miao and J. Liu, Strong averaging principle for slow-fast SPDEs with Poisson random measures, Discrete & Continuous Dynamical Systems-Series B, 20 (2015), 2233-2256.  doi: 10.3934/dcdsb.2015.20.2233. [34] J. Xu and Y. Miao, Lp (p > 2)-strong convergence of an averaging principle for two-time-scales jump-diffusion stochastic differential equations, Nonlinear Analysis: Hybrid Systems, 18 (2015), 33-47.  doi: 10.1016/j.nahs.2015.05.001. [35] J. Xu, Lp-strong convergence of the averaging principle for slow-fast SPDEs with jumps, Journal of Mathematical Analysis and Applications, 445 (2017), 342-373.  doi: 10.1016/j.jmaa.2016.07.058. [36] Y. Xu, R. Guo, D. Liu, H. Zhang and J. Duan, Stochastic averaging principle for dynamical systems with fractional Brownian motion, Discrete & Continuous Dynamical Systems-Series B, 19 (2014), 1197-1212.  doi: 10.3934/dcdsb.2014.19.1197. [37] Y. Xu, B. Pei and R. Guo, Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion, Discrete & Continuous Dynamical Systems-Series B, 20 (2015), 2257-2267.  doi: 10.3934/dcdsb.2015.20.2257. [38] Y. Xu, B. Pei and J. Wu, Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, Stochastics & Dynamics, 17 (2017), 1750013, 16pp. doi: 10.1142/S0219493717500137. [39] Y. Xu, J. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Physica D: Nonlinear Phenomena, 240 (2011), 1395-1401.  doi: 10.1016/j.physd.2011.06.001.

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##### References:
 [1] E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes, The Annals of Probability, 29 (1999), 766-801.  doi: 10.1214/aop/1008956692. [2] J. Bao, G. Yin and C. Yuan, Two-time-scale stochastic partial differential equations driven by alpha-stable noise: averaging principles, Bernoulli, 23 (2017), 645-669.  doi: 10.3150/14-BEJ677. [3] J. Bao and C. Yuan, Numerical analysis for neutral SPDEs driven by α-stable processes, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 17 (2014), 1450031, 16pp. doi: 10.1142/S0219025714500313. [4] P. Bezandry, Existence of almost periodic solutions for semilinear stochastic evolution equations driven by fractional Brownian motion, Electronic Journal of Differential Equations, 156 (2012), 1-21. [5] C. Bréhier, Strong and weak orders in averaging for SPDEs, Stochastic Processes and their Applications, 122 (2012), 2553-2593.  doi: 10.1016/j.spa.2012.04.007. [6] T. Caraballo, M. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047. [7] S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction–diffusion equations, Probability Theory and Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z. [8] P. Chow, Stochastic Partial Differential Equations, Second edition. Advances in Applied Mathematics. CRC Press, Boca Raton, FL, 2015. [9] I. Chueshov and B. Schmalfuss, Averaging of attractors and inertial manifolds for parabolic PDE with random coefficients, Advanced Nonlinear Studies, 5 (2005), 461-492.  doi: 10.1515/ans-2005-0402. [10] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513. [11] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, Amsterdam, 2014. [12] T. Duncan, Y. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion i. theory, SIAM Journal on Control and Optimization, 38 (2000), 582-612.  doi: 10.1137/S036301299834171X. [13] M. Freidlin and A. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4684-0176-9. [14] H. Fu and J. Liu, Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations, Journal of Mathematical Analysis and Applications, 384 (2011), 70-86.  doi: 10.1016/j.jmaa.2011.02.076. [15] H. Fu, L. Wan and J. Liu, Strong convergence in averaging principle for stochastic hyperbolic–parabolic equations with two time-scales, Stochastic Processes and their Applications, 125 (2015), 3255-3279.  doi: 10.1016/j.spa.2015.03.004. [16] D. Givon, Strong convergence rate for two-time-scale jump-diffusion stochastic differential sysytems, Multiscale Modeling & Simulation, 6 (2007), 577-594.  doi: 10.1137/060673345. [17] R. Guo and B. Pei, Stochastic averaging principles for multi-valued stochastic differential equations driven by Poisson point processes,, Stochastic Analysis and Applications, 36 (2018), 751-766.  doi: 10.1080/07362994.2018.1461567. [18] Y. Hu and B. Øksendal, Fractional white noise calculus and applications to finance, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 6 (2003), 1-32.  doi: 10.1142/S0219025703001110. [19] H. Hurst, Long-term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers, 116 (1951), 770-808. [20] J. Luo and T. Taniguchi, The existence and uniqueness for non-lipschitz stochastic neutral delay evolution equations driven by poisson jumps, Stochastics & Dynamics, 9 (2010), 135-152.  doi: 10.1142/S0219493709002592. [21] B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, Journal of Functional Analysis, 202 (2003), 277-305.  doi: 10.1016/S0022-1236(02)00065-4. [22] J. Mémin, Y. Mishura and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statistics & Probability Letters, 51 (2001), 197-206.  doi: 10.1016/S0167-7152(00)00157-7. [23] Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, Berlin, 2008. doi: 10.1007/978-3-540-75873-0. [24] D. Nualart, The Malliavin Calculus and Related Topics, second edition, Springer-Verlag, Berlin, 2006. [25] B. Øksendal, Stochastic Differential Equations, Springer, Heidelberg, 2003. [26] B. Pei, Y. Xu and J. Wu, Two-time-scales hyperbolic-parabolic equations driven by Poisson random measures: Existence, uniqueness and averaging principles,, Journal of Mathematical Analysis and Applications, 447 (2017), 243-268.  doi: 10.1016/j.jmaa.2016.10.010. [27] B. Pei, Y. Xu and G. Yin, Averaging principles for SPDEs driven by fractional Brownian motions with random delays modulated by two-time-scale Markov switching processes, Stochastics & Dynamics, 18 (2018), 1850023, 19pp. doi: 10.1142/S0219493718500235. [28] B. Pei, Y. Xu and G. Yin, Stochastic averaging for a class of two-time-scale systems of stochastic partial differential equations, Nonlinear Analysis: Theory Method and Application, 160 (2017), 159-176.  doi: 10.1016/j.na.2017.05.005. [29] B. Pei, Y. Xu, G. Yin and X. Zhang, Averaging principles for functional stochastic partial differential equations driven by a fractional Brownian motion modulated by two-time-scale Markovian switching processes, Nonlinear Analysis: Hybrid Systems, 27 (2018), 107-124.  doi: 10.1016/j.nahs.2017.08.008. [30] J. Seidler, Da Prato-Zabczyk's maximal inequality revisited I, Mathematica Bohemica, 118 (1993), 67-106. [31] W. Thompson, R. Kuske and A. Monahan, Stochastic averaging of dynamical systems with multiple time scales forced with α-stable noise, Multiscale Modeling & Simulation, 13 (2015), 1194-1223.  doi: 10.1137/140990632. [32] S. Tindel, C. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probability Theory and Related Fields, 127 (2003), 186-204.  doi: 10.1007/s00440-003-0282-2. [33] J. Xu, Y. Miao and J. Liu, Strong averaging principle for slow-fast SPDEs with Poisson random measures, Discrete & Continuous Dynamical Systems-Series B, 20 (2015), 2233-2256.  doi: 10.3934/dcdsb.2015.20.2233. [34] J. Xu and Y. Miao, Lp (p > 2)-strong convergence of an averaging principle for two-time-scales jump-diffusion stochastic differential equations, Nonlinear Analysis: Hybrid Systems, 18 (2015), 33-47.  doi: 10.1016/j.nahs.2015.05.001. [35] J. Xu, Lp-strong convergence of the averaging principle for slow-fast SPDEs with jumps, Journal of Mathematical Analysis and Applications, 445 (2017), 342-373.  doi: 10.1016/j.jmaa.2016.07.058. [36] Y. Xu, R. Guo, D. Liu, H. Zhang and J. Duan, Stochastic averaging principle for dynamical systems with fractional Brownian motion, Discrete & Continuous Dynamical Systems-Series B, 19 (2014), 1197-1212.  doi: 10.3934/dcdsb.2014.19.1197. [37] Y. Xu, B. Pei and R. Guo, Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion, Discrete & Continuous Dynamical Systems-Series B, 20 (2015), 2257-2267.  doi: 10.3934/dcdsb.2015.20.2257. [38] Y. Xu, B. Pei and J. Wu, Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, Stochastics & Dynamics, 17 (2017), 1750013, 16pp. doi: 10.1142/S0219493717500137. [39] Y. Xu, J. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Physica D: Nonlinear Phenomena, 240 (2011), 1395-1401.  doi: 10.1016/j.physd.2011.06.001.
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