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doi: 10.3934/dcdsb.2022113
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Wong-Zakai approximations of stochastic lattice systems driven by long-range interactions and multiplicative white noises

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

*Corresponding author: Kenan Wu

Received  December 2021 Revised  March 2022 Early access June 2022

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

In this paper, we study the Wong-Zakai approximations of a stochastic lattice differential equation with long-range interactions and multiplicative white noise at each node. We first prove the existence and uniqueness of pullback random attractors for lattice system driven by multiplicative white noises as well as the corresponding Wong-Zakai approximate system. Then, we prove the convergence of solutions and the upper semicontinuity of random attractors for the Wong-Zakai approximate system as the size of approximation approaches zero.

Citation: Yiju Chen, Xiaohu Wang, Kenan Wu. Wong-Zakai approximations of stochastic lattice systems driven by long-range interactions and multiplicative white noises. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022113
References:
[1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. 
[2]

P. W. BatesX. Chen and A. J. J. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.

[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 for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[5]

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.

[6]

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

[7]

T. CaraballoP. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.

[8]

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

[9]

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

[10]

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), 2150050, 30 pp. doi: 10.1142/S0219493721500507.

[11]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156.  doi: 10.1109/81.222795.

[12] I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. 
[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. 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.

[16]

Z. GuoX. YanW. Wang and X. Liu, Approximate the dynamical behavior for stochastic systems by Wong-Zakai approaching, J. Math. Anal. Appl., 457 (2018), 214-232.  doi: 10.1016/j.jmaa.2017.08.004.

[17]

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.

[18]

X. HanP. E. Kloeden and B. Usman, Long term behavior of a random Hopfield neural lattice model, Commun. Pure Appl. Anal., 18 (2019), 809-824.  doi: 10.3934/cpaa.2019039.

[19]

X. HanP. E. Kloeden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.  doi: 10.1088/1361-6544/ab6813.

[20]

Y. Hong and C. Yang, Strong convergence for discrete nonlinear Schrödinger equations in the continuum limit, SIAM J. Math. Anal., 51 (2019), 1297-1320.  doi: 10.1137/18M120703X.

[21]

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

[22]

D. F. Lawden, Elliptic Functions and Applications, Applied Mathematical Sciences, 80. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-3980-0.

[23]

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.

[24]

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

[25]

K. Lu and Q. Wang, Chaotic behavior in differential equations driven by a Brownian motion, J. Differential Equations, 251 (2011), 2853-2895.  doi: 10.1016/j.jde.2011.05.032.

[26]

S. F. MingaleevP. L. ChristiansenYu. B. GaidideiM. Johansson and K. Ø. Rasmussen, Models for energy and charge transport and storage in biomolecules, J. Biol. Phys., 25 (1999), 41-63.  doi: 10.1023/A:1005152704984.

[27]

J. M. Pereira, Global attractor for a generalized discrete nonlinear Schrödinger equation, Acta Appl. Math., 134 (2014), 173-183.  doi: 10.1007/s10440-014-9877-0.

[28]

W. M. Schouten-Straatman and H. J. Hupkes, Travelling wave solutions for fully discrete FitzHugh-Nagumo type equations with infinite-range interactions, J. Math. Anal. Appl., 502 (2021), 125272, 41 pp. doi: 10.1016/j.jmaa.2021.125272.

[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. ShenK. Lu and W. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differential Equations, 255 (2013), 4185-4225.  doi: 10.1016/j.jde.2013.08.003.

[31]

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

[32]

M. SuiY. WangX. Han and P. E. Kloeden, Random recurrent neural networks with delays, J. Differential Equations, 269 (2020), 8597-8639.  doi: 10.1016/j.jde.2020.06.008.

[33]

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

[34]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.

[35]

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.

[36]

X. WangP. E. Kloeden and X. Han, Attractors of Hopfield-type lattice models with increasing neuronal input, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 799-813.  doi: 10.3934/dcdsb.2019268.

[37]

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

[38]

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

[39]

X. Wang, K. Lu and B. Wang, Stationary approximations of stochastic wave equations on unbounded domains with critical exponents, J. Math. Phys., 62 (2021), 092702, 35 pp. doi: 10.1063/5.0011987.

[40]

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

[41]

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.

[42]

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.

[43]

X. YanX. Liu and M. Yang, Random attractors of stochastic partial differential equations: A smooth approximation approach, Stoch. Anal. Appl., 35 (2017), 1007-1029.  doi: 10.1080/07362994.2017.1345317.

[44]

S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368.  doi: 10.1016/j.jde.2004.02.005.

[45]

S. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients, J. Math. Anal. Appl., 395 (2012), 42-55.  doi: 10.1016/j.jmaa.2012.04.080.

show all references

References:
[1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. 
[2]

P. W. BatesX. Chen and A. J. J. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.

[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 for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[5]

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.

[6]

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

[7]

T. CaraballoP. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.

[8]

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

[9]

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

[10]

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), 2150050, 30 pp. doi: 10.1142/S0219493721500507.

[11]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156.  doi: 10.1109/81.222795.

[12] I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. 
[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. 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.

[16]

Z. GuoX. YanW. Wang and X. Liu, Approximate the dynamical behavior for stochastic systems by Wong-Zakai approaching, J. Math. Anal. Appl., 457 (2018), 214-232.  doi: 10.1016/j.jmaa.2017.08.004.

[17]

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.

[18]

X. HanP. E. Kloeden and B. Usman, Long term behavior of a random Hopfield neural lattice model, Commun. Pure Appl. Anal., 18 (2019), 809-824.  doi: 10.3934/cpaa.2019039.

[19]

X. HanP. E. Kloeden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.  doi: 10.1088/1361-6544/ab6813.

[20]

Y. Hong and C. Yang, Strong convergence for discrete nonlinear Schrödinger equations in the continuum limit, SIAM J. Math. Anal., 51 (2019), 1297-1320.  doi: 10.1137/18M120703X.

[21]

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

[22]

D. F. Lawden, Elliptic Functions and Applications, Applied Mathematical Sciences, 80. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-3980-0.

[23]

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.

[24]

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

[25]

K. Lu and Q. Wang, Chaotic behavior in differential equations driven by a Brownian motion, J. Differential Equations, 251 (2011), 2853-2895.  doi: 10.1016/j.jde.2011.05.032.

[26]

S. F. MingaleevP. L. ChristiansenYu. B. GaidideiM. Johansson and K. Ø. Rasmussen, Models for energy and charge transport and storage in biomolecules, J. Biol. Phys., 25 (1999), 41-63.  doi: 10.1023/A:1005152704984.

[27]

J. M. Pereira, Global attractor for a generalized discrete nonlinear Schrödinger equation, Acta Appl. Math., 134 (2014), 173-183.  doi: 10.1007/s10440-014-9877-0.

[28]

W. M. Schouten-Straatman and H. J. Hupkes, Travelling wave solutions for fully discrete FitzHugh-Nagumo type equations with infinite-range interactions, J. Math. Anal. Appl., 502 (2021), 125272, 41 pp. doi: 10.1016/j.jmaa.2021.125272.

[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. ShenK. Lu and W. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differential Equations, 255 (2013), 4185-4225.  doi: 10.1016/j.jde.2013.08.003.

[31]

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

[32]

M. SuiY. WangX. Han and P. E. Kloeden, Random recurrent neural networks with delays, J. Differential Equations, 269 (2020), 8597-8639.  doi: 10.1016/j.jde.2020.06.008.

[33]

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

[34]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.

[35]

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.

[36]

X. WangP. E. Kloeden and X. Han, Attractors of Hopfield-type lattice models with increasing neuronal input, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 799-813.  doi: 10.3934/dcdsb.2019268.

[37]

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

[38]

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

[39]

X. Wang, K. Lu and B. Wang, Stationary approximations of stochastic wave equations on unbounded domains with critical exponents, J. Math. Phys., 62 (2021), 092702, 35 pp. doi: 10.1063/5.0011987.

[40]

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

[41]

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.

[42]

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.

[43]

X. YanX. Liu and M. Yang, Random attractors of stochastic partial differential equations: A smooth approximation approach, Stoch. Anal. Appl., 35 (2017), 1007-1029.  doi: 10.1080/07362994.2017.1345317.

[44]

S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368.  doi: 10.1016/j.jde.2004.02.005.

[45]

S. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients, J. Math. Anal. Appl., 395 (2012), 42-55.  doi: 10.1016/j.jmaa.2012.04.080.

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