August  2022, 21(8): 2529-2560. doi: 10.3934/cpaa.2022059

Wong-Zakai approximations and pathwise dynamics of stochastic fractional lattice systems

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China

*Corresponding author

Received  November 2021 Revised  February 2022 Published  August 2022 Early access  March 2022

Fund Project: This work was supported by NSFC (11871049 and 12090013) and Young crop project of Sichuan University (2020SCUNL111)

This paper is concerned with the pathwise dynamics of stochastic fractional lattice systems driven by Wong-Zakai type approximation noises. The existence and uniqueness of pullback random attractor are established for the approximate system with a wide class of nonlinear diffusion term. For system with linear multiplicative noise and additive white noise, the upper semicontinuity of random attractors for the corresponding approximate system are also proved when the step size of the approximation approaches zero.

Citation: Yiju Chen, Xiaohu Wang, Kenan Wu. Wong-Zakai approximations and pathwise dynamics of stochastic fractional lattice systems. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2529-2560. doi: 10.3934/cpaa.2022059
References:
[1]

I. Anapolitanos, Remainder estimates for the long range behavior of the van der Waals interaction energy, Ann. H. Poincaré, 17 (2016), 1209-1261.  doi: 10.1007/s00023-015-0437-6.

[2]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[4]

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

[5]

P. C. Bressloff, Spatiotemporal dynamics of continuum neural fields, J. Phys. A, 45 (2012), 109 pp. doi: 10.1088/1751-8113/45/3/033001.

[6]

T. CaraballoX. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[7]

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

[8]

Y. Chen, C. Guo and X. Wang, Wong-Zakai approximations of second-order stochastic lattice systems driven by additive white noise, Stoch. Dyn., 22 (2022), 30 pp. doi: 10.1142/S0219493721500507.

[9]

Y. Chen and X. Wang, Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 20 pp. doi: 10.3934/dcdsb.2021271.

[10]

Ó. CiaurriT. A. GillespieL. RoncalJ. L. Torrea and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132 (2017), 109-131.  doi: 10.1007/s11854-017-0015-6.

[11]

Ó. CiaurriC. LizamaL. Roncal and J. L. Varona, On a connection between the discrete fractional Laplacian and superdiffusion, Appl. Math. Lett., 49 (2015), 119-125.  doi: 10.1016/j.aml.2015.05.007.

[12]

Ó. Ciaurri and L. Roncal, Hardy's inequality for the fractional powers of a discrete Laplacian, J. Anal., 26 (2018), 211-225.  doi: 10.1007/s41478-018-0141-2.

[13]

Ó. CiaurriL. RoncalP. R. StingaJ. L. Torrea and J. L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math., 330 (2018), 688-738.  doi: 10.1016/j.aim.2018.03.023.

[14]

A. Gu, Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5737-5767.  doi: 10.3934/dcdsb.2019104.

[15]

A. GuB. Guo and B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2495-2532.  doi: 10.3934/dcdsb.2020020.

[16]

A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst., 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.

[17]

A. Gu and B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.

[18]

A. Gu and B. Wang, Random attractors of reaction-diffusion equations without uniqueness driven by nonlinear colored noise, J. Math. Anal. Appl., 486 (2020), 23 pp. doi: 10.1016/j.jmaa.2020.123880.

[19]

X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[20]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differ. Equ., 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[21]

Z. HeX. ZhangT. Jiang and X. Liu, A Wong-Zakai approximation for random slow manifolds with application to parameter estimation, Nonlinear Dynam., 98 (2019), 403-426. 

[22]

T. Jiang, X. Liu and J. Duan, A Wong-Zakai approximation for random invariant manifolds, J. Math. Phys., 58 (2017), 19 pp. doi: 10.1063/1.5017932.

[23]

K. KirkpatrickE. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Commun. Math. Phys., 317 (2013), 563-591.  doi: 10.1007/s00220-012-1621-x.

[24]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[25]

C. Lizama and L. Roncal, Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 1365-1403.  doi: 10.3934/dcds.2018056.

[26]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differ. Equ., 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.

[27]

C. Martínez and M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Math. Studies 187, Amsterdam, 2001.

[28]

J. Shen and K. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differ. Equ., 263 (2017), 4929-4977.  doi: 10.1016/j.jde.2017.06.005.

[29]

J. ShenK. Lu and B. Wang, Convergence and center manifolds for differential equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4797-4840.  doi: 10.3934/dcds.2019196.

[30]

J. ShenJ. ZhaoK. Lu and B. Wang, The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differ. Equ., 266 (2019), 4568-4623.  doi: 10.1016/j.jde.2018.10.008.

[31]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Commun. Partial Differ. Equ., 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.

[32]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 31 pp. doi: 10.1142/S0219493714500099.

[33]

R. WangY. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

[34]

X. WangD. Li and J. Shen, Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2829-2855.  doi: 10.3934/dcdsb.2020207.

[35]

X. WangS. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.

[36]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differ. Equ., 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[37]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.

[38]

X. WangJ. ShenK. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differ. Equ., 280 (2021), 477-516.  doi: 10.1016/j.jde.2021.01.026.

[39]

E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Internat. J. Engrg. Sci., 3 (1965), 213-229.  doi: 10.1016/0020-7225(65)90045-5.

[40]

E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.

[41]

Y. Yang, J. Shu and X. Wang, Wong-Zakai approximations and random attractors of non-autonomous stochastic discrete complex Ginzburg-Landau equations, J. Math. Phys., 62 (2021), 29 pp. doi: 10.1063/5.0016914.

show all references

References:
[1]

I. Anapolitanos, Remainder estimates for the long range behavior of the van der Waals interaction energy, Ann. H. Poincaré, 17 (2016), 1209-1261.  doi: 10.1007/s00023-015-0437-6.

[2]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[4]

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

[5]

P. C. Bressloff, Spatiotemporal dynamics of continuum neural fields, J. Phys. A, 45 (2012), 109 pp. doi: 10.1088/1751-8113/45/3/033001.

[6]

T. CaraballoX. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[7]

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

[8]

Y. Chen, C. Guo and X. Wang, Wong-Zakai approximations of second-order stochastic lattice systems driven by additive white noise, Stoch. Dyn., 22 (2022), 30 pp. doi: 10.1142/S0219493721500507.

[9]

Y. Chen and X. Wang, Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 20 pp. doi: 10.3934/dcdsb.2021271.

[10]

Ó. CiaurriT. A. GillespieL. RoncalJ. L. Torrea and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132 (2017), 109-131.  doi: 10.1007/s11854-017-0015-6.

[11]

Ó. CiaurriC. LizamaL. Roncal and J. L. Varona, On a connection between the discrete fractional Laplacian and superdiffusion, Appl. Math. Lett., 49 (2015), 119-125.  doi: 10.1016/j.aml.2015.05.007.

[12]

Ó. Ciaurri and L. Roncal, Hardy's inequality for the fractional powers of a discrete Laplacian, J. Anal., 26 (2018), 211-225.  doi: 10.1007/s41478-018-0141-2.

[13]

Ó. CiaurriL. RoncalP. R. StingaJ. L. Torrea and J. L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math., 330 (2018), 688-738.  doi: 10.1016/j.aim.2018.03.023.

[14]

A. Gu, Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5737-5767.  doi: 10.3934/dcdsb.2019104.

[15]

A. GuB. Guo and B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2495-2532.  doi: 10.3934/dcdsb.2020020.

[16]

A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst., 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.

[17]

A. Gu and B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.

[18]

A. Gu and B. Wang, Random attractors of reaction-diffusion equations without uniqueness driven by nonlinear colored noise, J. Math. Anal. Appl., 486 (2020), 23 pp. doi: 10.1016/j.jmaa.2020.123880.

[19]

X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[20]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differ. Equ., 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[21]

Z. HeX. ZhangT. Jiang and X. Liu, A Wong-Zakai approximation for random slow manifolds with application to parameter estimation, Nonlinear Dynam., 98 (2019), 403-426. 

[22]

T. Jiang, X. Liu and J. Duan, A Wong-Zakai approximation for random invariant manifolds, J. Math. Phys., 58 (2017), 19 pp. doi: 10.1063/1.5017932.

[23]

K. KirkpatrickE. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Commun. Math. Phys., 317 (2013), 563-591.  doi: 10.1007/s00220-012-1621-x.

[24]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[25]

C. Lizama and L. Roncal, Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 1365-1403.  doi: 10.3934/dcds.2018056.

[26]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differ. Equ., 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.

[27]

C. Martínez and M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Math. Studies 187, Amsterdam, 2001.

[28]

J. Shen and K. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differ. Equ., 263 (2017), 4929-4977.  doi: 10.1016/j.jde.2017.06.005.

[29]

J. ShenK. Lu and B. Wang, Convergence and center manifolds for differential equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4797-4840.  doi: 10.3934/dcds.2019196.

[30]

J. ShenJ. ZhaoK. Lu and B. Wang, The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differ. Equ., 266 (2019), 4568-4623.  doi: 10.1016/j.jde.2018.10.008.

[31]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Commun. Partial Differ. Equ., 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.

[32]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 31 pp. doi: 10.1142/S0219493714500099.

[33]

R. WangY. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

[34]

X. WangD. Li and J. Shen, Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2829-2855.  doi: 10.3934/dcdsb.2020207.

[35]

X. WangS. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.

[36]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differ. Equ., 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[37]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.

[38]

X. WangJ. ShenK. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differ. Equ., 280 (2021), 477-516.  doi: 10.1016/j.jde.2021.01.026.

[39]

E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Internat. J. Engrg. Sci., 3 (1965), 213-229.  doi: 10.1016/0020-7225(65)90045-5.

[40]

E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.

[41]

Y. Yang, J. Shu and X. Wang, Wong-Zakai approximations and random attractors of non-autonomous stochastic discrete complex Ginzburg-Landau equations, J. Math. Phys., 62 (2021), 29 pp. doi: 10.1063/5.0016914.

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