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doi: 10.3934/dcdsb.2020226

Invariant measures of stochastic delay lattice systems

1. 

School of Mathematics, Shandong University, Jinan 250100, China

2. 

School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong 264005, China

3. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author: Bixiang Wang

Received  January 2020 Revised  May 2020 Published  July 2020

Fund Project: Zhang Chen is partially supported by NNSF of China grant 11471190, 11971260, NSF of Shandong Province grant ZR2014AM002. Xiliang Li is partially supported by NNSF of China grant 11971273 and NSF of Shandong Province grant ZR2018MA004

This paper is concerned with the existence and uniqueness of invariant measures for infinite-dimensional stochastic delay lattice systems defined on the entire integer set. For Lipschitz drift and diffusion terms, we prove the existence of invariant measures of the systems by showing the tightness of a family of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem. We also show the uniqueness of invariant measures when the Lipschitz coefficients of the nonlinear drift and diffusion terms are sufficiently small.

Citation: Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020226
References:
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V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 631-637.  doi: 10.1142/S0218127494000459.  Google Scholar

[2]

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

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P. W. Bates and A. Chmaj, On a discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.  Google Scholar

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P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, International J. Bifur. Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.  Google Scholar

[5]

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

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

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J. Bell and C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quarterly Appl. Math., 42 (1984), 1-14.  doi: 10.1090/qam/736501.  Google Scholar

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W. J. Beyn and S. Y. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dyn. Differential Equations, 15 (2003), 485-515.  doi: 10.1023/B:JODY.0000009745.41889.30.  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. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.  Google Scholar

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J. 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

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M. J. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stochastics and Dynamics, 11 (2011), 369-388.  doi: 10.1142/S0219493711003358.  Google Scholar

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show all references

References:
[1]

V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 631-637.  doi: 10.1142/S0218127494000459.  Google Scholar

[2]

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

[3]

P. W. Bates and A. Chmaj, On a discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.  Google Scholar

[4]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, International J. Bifur. Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.  Google Scholar

[5]

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

[6]

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

[7]

J. Bell and C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quarterly Appl. Math., 42 (1984), 1-14.  doi: 10.1090/qam/736501.  Google Scholar

[8]

W. J. Beyn and S. Y. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dyn. Differential Equations, 15 (2003), 485-515.  doi: 10.1023/B:JODY.0000009745.41889.30.  Google Scholar

[9]

Z. BrzezniakM. Ondrejat 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

[10]

Z. BrzezniakE. Motyl and M. Ondrejat, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, Annals of Probability, 45 (2017), 3145-3201.  doi: 10.1214/16-AOP1133.  Google Scholar

[11]

O. Butkovsky and M. Scheutzow, Invariant measures for stochastic functional differential equations, Electron. J. Probab., 22 (2017), 1-23.  doi: 10.1214/17-EJP122.  Google Scholar

[12]

T. CaraballoM. J. Garrido-Atienza and B. Schmalfuss, Exponential stability of stationary solutions for semilinear stochastic evolution equations with delays, Discret. Contin. Dyn. Syst., 18 (2007), 271-293.  doi: 10.3934/dcds.2007.18.271.  Google Scholar

[13]

T. CaraballoM. 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

[14]

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

[15]

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

[16]

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

[17]

S. N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 49 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.  Google Scholar

[18]

S. N. ChowJ. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Computational Dynamics, 4 (1996), 109-178.   Google Scholar

[19]

S. N. Chow and W. Shen, Dynamics in a discrete Nagumo equation: Spatial topological chaos, SIAM J. Appl. Math., 55 (1995), 1764-1781.  doi: 10.1137/S0036139994261757.  Google Scholar

[20]

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

[21]

L. O. Chua and Y. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Systems, 35 (1988), 1257-1272.  doi: 10.1109/31.7600.  Google Scholar

[22] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[23]

J. 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

[24]

C. E. Elmer and E. S. Van Vleck, Analysis and computation of traveling wave solutions of bistable differential-difference equations, Nonlinearity, 12 (1999), 771-798.  doi: 10.1088/0951-7715/12/4/303.  Google Scholar

[25]

C. E. Elmer and E. S. Van Vleck, Traveling waves solutions for bistable differential-difference equations with periodic diffusion, SIAM J. Appl. Math., 61 (2001), 1648-1679.  doi: 10.1137/S0036139999357113.  Google Scholar

[26]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I.  Google Scholar

[27]

A. Es-SarhirM. Scheutzow and O. van Gaans, Invariant measures for stochastic functional differential equations with superlinear drift term, Differential Integral Equations, 23 (2010), 189-200.   Google Scholar

[28]

M. J. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stochastics and Dynamics, 11 (2011), 369-388.  doi: 10.1142/S0219493711003358.  Google Scholar

[29]

K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.  Google Scholar

[30]

J. K. Hale, Functional differential equations with infinite delays, J. Math. Anal. Appl., 48 (1974), 276-283.  doi: 10.1016/0022-247X(74)90233-9.  Google Scholar

[31]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[32]

X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493.  doi: 10.1016/j.jmaa.2010.11.032.  Google Scholar

[33]

X. HanW. Shen and S. 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

[34]

X. 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

[35]

X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $Z^k$ in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254.  doi: 10.1016/j.jmaa.2012.07.015.  Google Scholar

[36]

Y. Hino, T. Naito and S. Murakami, Functional Differential Equations with Infinite Delay, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.  Google Scholar

[37]

K. Ito and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75.  doi: 10.1215/kjm/1250524705.  Google Scholar

[38]

R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578.  Google Scholar

[39]

N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrodinger equation, J. Differential Equations, 217 (2005), 88-123.  doi: 10.1016/j.jde.2005.06.002.  Google Scholar

[40]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038.  Google Scholar

[41]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82.  doi: 10.1016/S0022-5193(05)80465-5.  Google Scholar

[42]

J. Kim, On the stochastic Burgers equation with polynomial nonlinearity in the real line, Discrete Continuous Dynam. Systems - B, 6 (2006), 835-866.  doi: 10.3934/dcdsb.2006.6.835.  Google Scholar

[43]

J. Kim, On the stochastic Benjamin-Ono equation, J. Differential Equations, 228 (2006), 737-768.  doi: 10.1016/j.jde.2005.11.005.  Google Scholar

[44]

J. Kim, Periodic and invariant measures for stochastic wave equations, Electronic Journal of Differential Equations, 2004 (2004), 1-30.   Google Scholar

[45]

J. Kim, Invariant measures for a stochastic nonlinear Schrodinger equation, Indiana University Mathematics Journal, 55 (2006), 687-717.  doi: 10.1512/iumj.2006.55.2701.  Google Scholar

[46] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986.   Google Scholar
[47] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, Boston, 1993.   Google Scholar
[48]

Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Volterra type system, J. Differential Equations, 103 (1993), 221-246.  doi: 10.1006/jdeq.1993.1048.  Google Scholar

[49]

X. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Woodhead Publishing Limited, Chichester, 2008.  Google Scholar

[50]

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