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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 & 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.  Google Scholar

[2]

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

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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.  Google Scholar

[12]

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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

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