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February  2020, 19(2): 811-834. doi: 10.3934/cpaa.2020038

Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise

Caibin Zeng 1, , Xiaofang Lin 1, , Jianhua Huang 2, and  Qigui Yang 1,,

1. 

School of Mathematics, South China University of Technology, Guangzhou 510640, China
2. 

College of Science, National University of Defense Technology, Changsha 410073, China

* Corresponding author

Received  November 2018 Revised  June 2019 Published  October 2019

The fBm-driving rough stochastic lattice dynamical system with a general diffusion term is investigated. First, an area element in space of tensor is desired to define the rough path integral using the Chen-equality and fractional calculus. Under certain conditions, the considered equation is proved to possess a unique local mild path-area solution.

Keywords: Stochastic lattice system, fractional Brownian motion, pathwise solution, Chen-equality.
Mathematics Subject Classification: Primary: 60H15, 60G22; Secondary: 37K60, 37HXX.
Citation: Caibin Zeng, Xiaofang Lin, Jianhua Huang, Qigui Yang. Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise. Communications on Pure and Applied Analysis, 2020, 19 (2) : 811-834. doi: 10.3934/cpaa.2020038
References:
[1]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics and Dynamics, 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[2]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internal Journal of Bifurcation and Chaos, 11 (2001) 143–153. doi: 10.1142/S021812740100203.

[3]

P. W. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D: Nonlinear Phenomena, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[4]

H. Bessaih, M. J. Garrido-Atienza, X. Han and B. Schmalfuss, Stochastic lattice dynamical systems with fractional noise, SIAM Journal on Mathematical Analysis, 49 (2017), 1495-1518.  doi: 10.1137/16M1085504.

[5]

T. Caraballo, X. Han, B. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Analysis-theory Methods & Applications, 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[6]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Frontiers of Mathematics in China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.

[7]

T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, Journal of Differential Equations and Applications, 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010.

[8]

T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, Journal of Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[9]

B. Chen, Some pinching and classification theorems for minimal submanifolds, Archiv der Mathematik, 60 (1993), 568-578.  doi: 10.1007/BF01236084.

[10]

Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuss, Pathwise solutions of SPDEs driven by hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete and Continuous Dynamical Systems, 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79.

[11]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems-I, IEEE Transactions on Circuits and Systems, 42 (1995), 746-751.  doi: 10.1109/81.473583.

[12]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, New York, 2003.

[13]

M. Garrido-Atienza, K. Lu and B. Schmalfuss, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3, 1/2]$, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), pp. 2553–2581. doi: 10.3934/dcdsb.2015.20.2553.

[14]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in (1/3, 1/2]$, SIAM Journal on Applied Dynamical Systems, 15 (2016), 625-654.  doi: 10.1137/15M1030303.

[15]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Lévy-areas of Ornstein-Uhlenbeck processes in Hilbert-spaces, in Continuous and Distributed Systems II: Theory and Applications (eds. V.A. Sadovnichiy and M.Z. Zgurovsky) doi: 10.1007/978-3-319-19075-4_10.

[16]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Pathwise solutions to stochastic partial differential equations driven by fractional Brownian motions with Hurst parameters in $(1/3, 1/2]$, preprint, arXiv: 1205.6735v3.

[17]

M. J. Garrido-Atienza, A. Neuenkirch and B. Schmalfus, Asymptotical stability of differential equations driven by Hölder continuous paths, Journal of Dynamics and Differential Equations, 30 (2018), 1-19.  doi: 10.1007/s10884-017-9574-6.

[18]

M. J. Garrido-Atienza and B. Schmalfuß, Ergodicity of the infinite dimensional fractional Brownian motion, Journal of Dynamics and Differential Equations, 23 (2011), pp. 671–681. doi: 10.1007/s10884-011-9222-5.

[19]

A. Gu, Random attractors of stochastic lattice dynamical systems driven by fractional Brownian motions, International Journal of Bifurcation and Chaos, 23 (2013), 1350041. doi: 10.1142/S0218127413500417.

[20]

A. Gu and Y. Li, Singleton sets random attractor for stochastic FitzHugh-Nagumo lattice equations driven by fractional Brownian motions, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3929-3937.  doi: 10.1016/j.cnsns.2014.04.005.

[21]

A. Gu, C. Zeng and Y. Li, Synchronization of systems with fractional environmental noises on finite lattice, Fractional Calculus and Applied Analysis, 18 (2015), 891-910.  doi: 10.1515/fca-2015-0054.

[22]

X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, Journal of Mathematical Analysis and Applications, 376 (2011), 481-493.  doi: 10.1016/j.jmaa.2010.11.032.

[23]

X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, Journal of Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[24]

Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Transactions of the American Mathematical Society, 361 (2009), 2689-2718.  doi: 10.1090/S0002-9947-08-04631-X.

[25]

J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Physica D: Nonlinear Phenomena, 233 (2007), 83-94.  doi: 10.1016/j.physd.2007.06.008.

[26]

Y. Lv and J. Sun, Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D: Nonlinear Phenomena, 221 (2006), 157-169.  doi: 10.1016/j.physd.2006.07.02.

[27]

Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems, Chaos, Solitons & Fractals, 27 (2006), 1080-1090.  doi: 10.1016/j.chaos.2005.04.089.

[28]

J. Mallet-Paret and S.-N. Chow, Pattern formation and spatial chaos in lattice dynamical systems-II, IEEE Transactions on Circuits and Systems, 42 (1995), 752-756.  doi: 10.1109/81.473583.

[29]

S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[30]

B. Wang, Dynamics of systems on infinite lattices, Journal of Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.

[31]

X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.

[32]

Y. Wang, J. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Analysis-Theory Methods & Applications, 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020.

[33]

S. Zhou, Attractors for second order lattice dynamical systems, Journal of Differential Equations, 179 (2002), 606-624.  doi: 10.1006/jdeq.2001.4032.

[34]

S. Zhou, Attractors and approximations for lattice dynamical systems, Journal of Differential Equations, 200 (2004), 342-268.  doi: 10.1016/j.jde.2004.02.005.

[35]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, Journal of Differential Equations, 224 (2006), 172-204.  doi: 10.1016/j.jde.2005.06.024.

show all references

References:
[1]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics and Dynamics, 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[2]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internal Journal of Bifurcation and Chaos, 11 (2001) 143–153. doi: 10.1142/S021812740100203.

[3]

P. W. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D: Nonlinear Phenomena, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[4]

H. Bessaih, M. J. Garrido-Atienza, X. Han and B. Schmalfuss, Stochastic lattice dynamical systems with fractional noise, SIAM Journal on Mathematical Analysis, 49 (2017), 1495-1518.  doi: 10.1137/16M1085504.

[5]

T. Caraballo, X. Han, B. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Analysis-theory Methods & Applications, 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[6]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Frontiers of Mathematics in China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.

[7]

T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, Journal of Differential Equations and Applications, 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010.

[8]

T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, Journal of Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[9]

B. Chen, Some pinching and classification theorems for minimal submanifolds, Archiv der Mathematik, 60 (1993), 568-578.  doi: 10.1007/BF01236084.

[10]

Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuss, Pathwise solutions of SPDEs driven by hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete and Continuous Dynamical Systems, 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79.

[11]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems-I, IEEE Transactions on Circuits and Systems, 42 (1995), 746-751.  doi: 10.1109/81.473583.

[12]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, New York, 2003.

[13]

M. Garrido-Atienza, K. Lu and B. Schmalfuss, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3, 1/2]$, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), pp. 2553–2581. doi: 10.3934/dcdsb.2015.20.2553.

[14]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in (1/3, 1/2]$, SIAM Journal on Applied Dynamical Systems, 15 (2016), 625-654.  doi: 10.1137/15M1030303.

[15]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Lévy-areas of Ornstein-Uhlenbeck processes in Hilbert-spaces, in Continuous and Distributed Systems II: Theory and Applications (eds. V.A. Sadovnichiy and M.Z. Zgurovsky) doi: 10.1007/978-3-319-19075-4_10.

[16]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Pathwise solutions to stochastic partial differential equations driven by fractional Brownian motions with Hurst parameters in $(1/3, 1/2]$, preprint, arXiv: 1205.6735v3.

[17]

M. J. Garrido-Atienza, A. Neuenkirch and B. Schmalfus, Asymptotical stability of differential equations driven by Hölder continuous paths, Journal of Dynamics and Differential Equations, 30 (2018), 1-19.  doi: 10.1007/s10884-017-9574-6.

[18]

M. J. Garrido-Atienza and B. Schmalfuß, Ergodicity of the infinite dimensional fractional Brownian motion, Journal of Dynamics and Differential Equations, 23 (2011), pp. 671–681. doi: 10.1007/s10884-011-9222-5.

[19]

A. Gu, Random attractors of stochastic lattice dynamical systems driven by fractional Brownian motions, International Journal of Bifurcation and Chaos, 23 (2013), 1350041. doi: 10.1142/S0218127413500417.

[20]

A. Gu and Y. Li, Singleton sets random attractor for stochastic FitzHugh-Nagumo lattice equations driven by fractional Brownian motions, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3929-3937.  doi: 10.1016/j.cnsns.2014.04.005.

[21]

A. Gu, C. Zeng and Y. Li, Synchronization of systems with fractional environmental noises on finite lattice, Fractional Calculus and Applied Analysis, 18 (2015), 891-910.  doi: 10.1515/fca-2015-0054.

[22]

X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, Journal of Mathematical Analysis and Applications, 376 (2011), 481-493.  doi: 10.1016/j.jmaa.2010.11.032.

[23]

X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, Journal of Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[24]

Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Transactions of the American Mathematical Society, 361 (2009), 2689-2718.  doi: 10.1090/S0002-9947-08-04631-X.

[25]

J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Physica D: Nonlinear Phenomena, 233 (2007), 83-94.  doi: 10.1016/j.physd.2007.06.008.

[26]

Y. Lv and J. Sun, Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D: Nonlinear Phenomena, 221 (2006), 157-169.  doi: 10.1016/j.physd.2006.07.02.

[27]

Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems, Chaos, Solitons & Fractals, 27 (2006), 1080-1090.  doi: 10.1016/j.chaos.2005.04.089.

[28]

J. Mallet-Paret and S.-N. Chow, Pattern formation and spatial chaos in lattice dynamical systems-II, IEEE Transactions on Circuits and Systems, 42 (1995), 752-756.  doi: 10.1109/81.473583.

[29]

S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[30]

B. Wang, Dynamics of systems on infinite lattices, Journal of Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.

[31]

X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.

[32]

Y. Wang, J. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Analysis-Theory Methods & Applications, 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020.

[33]

S. Zhou, Attractors for second order lattice dynamical systems, Journal of Differential Equations, 179 (2002), 606-624.  doi: 10.1006/jdeq.2001.4032.

[34]

S. Zhou, Attractors and approximations for lattice dynamical systems, Journal of Differential Equations, 200 (2004), 342-268.  doi: 10.1016/j.jde.2004.02.005.

[35]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, Journal of Differential Equations, 224 (2006), 172-204.  doi: 10.1016/j.jde.2005.06.024.

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