Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise

Renhai Wang; Bixiang Wang

doi:10.3934/dcdsb.2020019

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July 2020, 25(7): 2461-2493. doi: 10.3934/dcdsb.2020019

Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise

Renhai Wang ^1,, and Bixiang Wang ^2,

1.	School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2.	Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

^* Corresponding author: rwang-math@outlook.com (Renhai Wang)

Received June 2019 Revised August 2019 Published July 2020 Early access April 2020

Fund Project: The first author is supported by the Innovation Project of Chongqing grant CYB18115 and the China Scholarship Council(CSC No.201806990064)

This paper is concerned with the global existence and random dynamics of non-autonomous stochastic second-order lattice systems driven by infinite-dimensional nonlinear noise defined on higher-dimensional integer sets. We first show the existence and uniqueness of mean square solutions to the equations when the nonlinear drift term has a polynomial growth of arbitrary order and the diffusion term is locally Lipschitz continuous. We then prove that the mean random dynamical system associated with the solution operator possesses a unique tempered weak pullback mean random attractor in a Bochner space under certain conditions. We finally establish the existence of invariant measures for the stochastic systems in $ \ell^2\times\ell^2 $ by showing the tightness of a family of distribution laws of solutions via the idea of uniform tail-estimates on the solutions.

Keywords: Invariant measure, weak mean random attractor, tightness, nonlinear noise, lattice wave equation.

Mathematics Subject Classification: Primary: 37L55; Secondary: 37B55, 35B41, 35B40.

Citation: Renhai Wang, Bixiang Wang. Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2461-2493. doi: 10.3934/dcdsb.2020019

References:

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[4]	Z. Brzeźniak, E. Motyl and M. Ondrejat, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, Ann. Probab., 45 (2017), 3145-3201. doi: 10.1214/16-AOP1133. Google Scholar
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[6]	T. Caraballo, X. Y. Han, B. 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. Google Scholar
[7]	T. Caraballo, F. Morillas and J. Valerom, 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. Google Scholar
[8]	T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455. doi: 10.3934/dcdsb.2010.14.439. Google Scholar
[9]	T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684. doi: 10.1016/j.na.2011.02.047. Google Scholar
[10]	S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, Ⅰ, Ⅱ, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746-751. doi: 10.1109/81.473583. Google Scholar
[11]	H. Y. Cui and P. E. Kloeden, Invariant forward attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186. doi: 10.1016/j.jde.2018.07.028. Google Scholar
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[19]	X. Y. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009. doi: 10.1016/j.jde.2016.05.015. Google Scholar
[20]	X. Y. Han, W. X. Shen and S. F. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018. Google Scholar
[21]	J. U. Kim, Periodic and invariant measures for stochastic wave equations, Electronic journal of differential equations, 2004 (2004), 30 pp. Google Scholar
[22]	J. U. Kim, On the stochastic Burgers equation with polynomial nonlinearity in the real line, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 835-866. doi: 10.3934/dcdsb.2006.6.835. Google Scholar
[23]	J. U. Kim, On the stochastic Benjamin-Ono equation, Journal of Differential Equations, 228 (2006), 737-768. doi: 10.1016/j.jde.2005.11.005. Google Scholar
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[27]	D. S. Li, K. N. Lu, B. X. Wang and X. H. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208. doi: 10.1016/j.jde.2016.10.024. Google Scholar
[28]	Y. R. Li, A. H. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021. Google Scholar
[29]	C. B. Li and J. C. Sprott, An infinite 3-D quasiperiodic lattice of chaotic attractors, Phys. Lett. A, 382 (2018), 581-587. doi: 10.1016/j.physleta.2017.12.022. Google Scholar
[30]	W.-W. Lin and Y.-Q. Wang, Proof of synchronized chaotic behaviors in coupled map lattices, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 1493-1500. doi: 10.1142/S0218127411029069. Google Scholar
[31]	K. N. Lu and B. X. 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. Google Scholar
[32]	O. Misiats, O. Stanzhytskyi and N. K. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, J. Theoret. Probab., 29 (2016), 996-1026. doi: 10.1007/s10959-015-0606-z. Google Scholar
[33]	X. H. Wang, K. N. Lu and B. X. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047. doi: 10.1137/140991819. Google Scholar
[34]	X. H. Wang, K. N. Lu and B. X. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2018), 378-424. doi: 10.1016/j.jde.2017.09.006. Google Scholar
[35]	B. X. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2. Google Scholar
[36]	B. X. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003. Google Scholar
[37]	B. X. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differ. Equ., 31 (2019), 2177–2204, https://doi.org/10.1007/s10884-018-9696-5. doi: 10.1007/s10884-018-9696-5. Google Scholar
[38]	B. X. Wang, Dynamics of stochastic reaction-diffusion lattice system driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104-132. doi: 10.1016/j.jmaa.2019.04.015. Google Scholar
[39]	B. X. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1–59, https://doi.org/10.1016/j.jde.2019.08.007. doi: 10.1016/j.jde.2019.08.007. Google Scholar
[40]	B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015. Google Scholar
[41]	B. X. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$, Tran. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5. Google Scholar
[42]	B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar
[43]	R. H. Wang, Y. R. Li and B. X. 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. Google Scholar
[44]	R. H. Wang and Y. R. Li, Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients, Appl. Math. Comput., 354 (2019), 86-102. doi: 10.1016/j.amc.2019.02.036. Google Scholar
[45]	Y. J. Wang, J. H. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal., 135 (2016), 205-222. doi: 10.1016/j.na.2016.01.020. Google Scholar
[46]	W. Q. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203. doi: 10.1016/j.jmaa.2017.06.025. Google Scholar
[47]	W. Q. Zhao and Y. J. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell^p_\rho$, Appl. Math. Comput., 291 (2016), 226-243. doi: 10.1016/j.amc.2016.06.045. Google Scholar
[48]	S. F. 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. Google Scholar
[49]	S. F. Zhou and X. Y. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dyn. Differ. Equ., 24 (2012), 601-631. doi: 10.1007/s10884-012-9260-7. Google Scholar
[50]	S. F. Zhou and Z. J. Wang, Finite fractal dimensions of random attractors for stochastic FitzHugh-Nagumo system with multiplicative white noise, J. Math. Anal. Appl., 441 (2016), 648-667. doi: 10.1016/j.jmaa.2016.04.038. Google Scholar
[51]	S. F. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36 (2017), 2887-2914. doi: 10.3934/dcds.2016.36.2887. Google Scholar

show all references

References:

[1]	L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Google Scholar
[2]	P. W. Bates, K. N. Lu and B. X. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017. Google Scholar
[3]	P. W. Bates, K. N. Lu and B. X. Wang, Tempered random attractors for parabolic equations in weighted spaces, J. Math. Phys., 54 (2013), 081505, 26 pp. doi: 10.1063/1.4817597. Google Scholar
[4]	Z. Brzeźniak, E. Motyl and M. Ondrejat, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, Ann. Probab., 45 (2017), 3145-3201. doi: 10.1214/16-AOP1133. Google Scholar
[5]	Z. Brzeźniak, M. Ondreját and J. Seidler, Invariant measures for stochastic nonlinear beam and wave equations, J. Differential Equations, 260 (2016), 4157-4179. doi: 10.1016/j.jde.2015.11.007. Google Scholar
[6]	T. Caraballo, X. Y. Han, B. 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. Google Scholar
[7]	T. Caraballo, F. Morillas and J. Valerom, 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. Google Scholar
[8]	T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455. doi: 10.3934/dcdsb.2010.14.439. Google Scholar
[9]	T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684. doi: 10.1016/j.na.2011.02.047. Google Scholar
[10]	S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, Ⅰ, Ⅱ, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746-751. doi: 10.1109/81.473583. Google Scholar
[11]	H. Y. Cui and P. E. Kloeden, Invariant forward attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186. doi: 10.1016/j.jde.2018.07.028. Google Scholar
[12]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Second edition. Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781107295513. Google Scholar
[13]	J.-P. Eckmann and M. Hairer, Invariant measures for stochastic partial differential equations in unbounded domains, Nonlinearity, 14 (2001), 133-151. doi: 10.1088/0951-7715/14/1/308. Google Scholar
[14]	T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction systems, Physica D, 67 (1993), 237-244. doi: 10.1016/0167-2789(93)90208-I. Google Scholar
[15]	A. H. Gu and Y. R. Li, Dynamic behavior of stochastic $p$-Laplacian-type lattice equations, Stoch. Dyn., 17 (2017), 1750040, 19 pp. doi: 10.1142/S021949371750040X. Google Scholar
[16]	J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations, 246 (2009), 3818-3833. doi: 10.1016/j.jde.2009.03.010. Google Scholar
[17]	X. Y. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dyn., 12 (2012), 1150024, 20 pp. doi: 10.1142/S0219493711500249. Google Scholar
[18]	X. Y. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Physica D, 389 (2019), 1-12. doi: 10.1016/j.physd.2018.09.004. Google Scholar
[19]	X. Y. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009. doi: 10.1016/j.jde.2016.05.015. Google Scholar
[20]	X. Y. Han, W. X. Shen and S. F. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018. Google Scholar
[21]	J. U. Kim, Periodic and invariant measures for stochastic wave equations, Electronic journal of differential equations, 2004 (2004), 30 pp. Google Scholar
[22]	J. U. Kim, On the stochastic Burgers equation with polynomial nonlinearity in the real line, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 835-866. doi: 10.3934/dcdsb.2006.6.835. Google Scholar
[23]	J. U. Kim, On the stochastic Benjamin-Ono equation, Journal of Differential Equations, 228 (2006), 737-768. doi: 10.1016/j.jde.2005.11.005. Google Scholar
[24]	P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438. doi: 10.1016/j.jde.2012.05.016. Google Scholar
[25]	P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. Google Scholar
[26]	D. S. Li, B. X. Wang and X. H. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602. doi: 10.1016/j.jde.2016.10.024. Google Scholar
[27]	D. S. Li, K. N. Lu, B. X. Wang and X. H. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208. doi: 10.1016/j.jde.2016.10.024. Google Scholar
[28]	Y. R. Li, A. H. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021. Google Scholar
[29]	C. B. Li and J. C. Sprott, An infinite 3-D quasiperiodic lattice of chaotic attractors, Phys. Lett. A, 382 (2018), 581-587. doi: 10.1016/j.physleta.2017.12.022. Google Scholar
[30]	W.-W. Lin and Y.-Q. Wang, Proof of synchronized chaotic behaviors in coupled map lattices, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 1493-1500. doi: 10.1142/S0218127411029069. Google Scholar
[31]	K. N. Lu and B. X. 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. Google Scholar
[32]	O. Misiats, O. Stanzhytskyi and N. K. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, J. Theoret. Probab., 29 (2016), 996-1026. doi: 10.1007/s10959-015-0606-z. Google Scholar
[33]	X. H. Wang, K. N. Lu and B. X. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047. doi: 10.1137/140991819. Google Scholar
[34]	X. H. Wang, K. N. Lu and B. X. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2018), 378-424. doi: 10.1016/j.jde.2017.09.006. Google Scholar
[35]	B. X. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2. Google Scholar
[36]	B. X. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003. Google Scholar
[37]	B. X. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differ. Equ., 31 (2019), 2177–2204, https://doi.org/10.1007/s10884-018-9696-5. doi: 10.1007/s10884-018-9696-5. Google Scholar
[38]	B. X. Wang, Dynamics of stochastic reaction-diffusion lattice system driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104-132. doi: 10.1016/j.jmaa.2019.04.015. Google Scholar
[39]	B. X. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1–59, https://doi.org/10.1016/j.jde.2019.08.007. doi: 10.1016/j.jde.2019.08.007. Google Scholar
[40]	B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015. Google Scholar
[41]	B. X. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$, Tran. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5. Google Scholar
[42]	B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar
[43]	R. H. Wang, Y. R. Li and B. X. 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. Google Scholar
[44]	R. H. Wang and Y. R. Li, Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients, Appl. Math. Comput., 354 (2019), 86-102. doi: 10.1016/j.amc.2019.02.036. Google Scholar
[45]	Y. J. Wang, J. H. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal., 135 (2016), 205-222. doi: 10.1016/j.na.2016.01.020. Google Scholar
[46]	W. Q. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203. doi: 10.1016/j.jmaa.2017.06.025. Google Scholar
[47]	W. Q. Zhao and Y. J. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell^p_\rho$, Appl. Math. Comput., 291 (2016), 226-243. doi: 10.1016/j.amc.2016.06.045. Google Scholar
[48]	S. F. 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. Google Scholar
[49]	S. F. Zhou and X. Y. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dyn. Differ. Equ., 24 (2012), 601-631. doi: 10.1007/s10884-012-9260-7. Google Scholar
[50]	S. F. Zhou and Z. J. Wang, Finite fractal dimensions of random attractors for stochastic FitzHugh-Nagumo system with multiplicative white noise, J. Math. Anal. Appl., 441 (2016), 648-667. doi: 10.1016/j.jmaa.2016.04.038. Google Scholar
[51]	S. F. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36 (2017), 2887-2914. doi: 10.3934/dcds.2016.36.2887. Google Scholar

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American Institute of Mathematical Sciences

Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise

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Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise

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