Exponential Stability of Non-Autonomous Stochastic Delay Lattice Systems with Multiplicative Noise

Arnold, L.: Random Dynamical System. Springer, New York (1998)

Bates, P.W., Chmaj, A.: On a discrete convolution model for phase transitions. Arch. Ration. Mech. Anal. 150, 281–305 (1999)

Bates, P.W., Lu, K., Wang, B.: Attractors for lattice dynamical systems. Int. J. Bifur. Chaos 11, 143–153 (2001)

Bates, P.W., Chen, X., Chmaj, A.: Traveling waves of bistable dynamics on a lattice. SIAM J. Math. Anal. 35, 520–546 (2003)

Bates, P.W., Lisei, H., Lu, K.: Attractors for stochastic lattice dynamical systems. Stoch. Dynam. 6, 1–21 (2006)

Bates, P.W., Lu, K., Wang, B.: Attractors for non-autonomous stochastic lattice systems in weighted space. Physica D 289, 32–50 (2014)

Beyn, W.J., Pilyugin, S.Y.: Attractors of reaction diffusion systems on infinite lattices. J. Dynam. Differ. Equ. 15, 485–515 (2003)

Caraballo, T., Lu, K.: Attractors for stochastic lattice dynamical systems with a multiplicative noise. Front. Math. China 3, 317–335 (2008)

Caraballo, T., Kloeden, P., Schmalfuss, B.: Exponentially stable stationary solutions for stochastic evolutions equations and their perturbations. Appl. Math. Optim. 50, 183–207 (2004)

Caraballo, T., Garrido-Atienza, M.J., Schmalfuss, B.: Exponential stability of stationary solutions for semilinear stochastic evolution equations with delays. Discret. Contin. Dyn. Syst. 18, 271–293 (2007)

Caraballo, T., Garrido-Atienza, M.J., Schmalfuss, B., Valero, J.: Non-autonomous and random attractors for delay random semilinear equations without uniqueness. Discret. Contin. Dyn. Syst. 21, 415–433 (2008)

Caraballo, T., Morillas, F., Valero, J.: Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity. J. Differ. Equ. 253, 667–693 (2012)

Chow, S.N., Mallet-Paret, J.: Pattern formation and spatial chaos in lattice dynamical systems, I. IEEE Trans. Circuits Syst. 42, 746–751 (1995)

Chow, S.N., Shen, W.: Dynamics in a discrete Nagumo equation: spatial topological chaos. SIAM J. Appl. Math. 55, 1764–1781 (1995)

Chow, S.N., Mallet-Paret, J., Van Vleck, E.S.: Pattern formation and spatial chaos in spatially discrete evolution equations. Random Comput. Dynam. 4, 109–178 (1996)

Chow, S.N., Mallet-Paret, J., Shen, W.: Traveling waves in lattice dynamical systems. J. Differ. Equ. 49, 248–291 (1998)

Chua, L.O., Roska, T.: The CNN paradigm. IEEE Trans. Circuits Syst. 40, 147–156 (1993)

Crauel, H., Flandoli, F.: Attractor for random dynamical systems. Probab. Theory Relat. Fields 100, 365–393 (1994)

Ding, X., Jiang, J.: Random attractors for stochastic retarded lattice dynamical systems. Abstr. Appl. Anal. 2012, 409282 (2012)

Elmer, C.E., Van Vleck, E.S.: Analysis and computation of traveling wave solutions of bistable differential–difference equations. Nonlinearity 12, 771–798 (1999)

Erneux, T., Nicolis, G.: Propagating waves in discrete bistable reaction diffusion systems. Physica D 67, 237–244 (1993)

Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative noise. Stoch. Stoch. Rep. 59, 21–45 (1996)

Han, X., Shen, W., Zhou, S.: Random attractors for stochastic lattice dynamical systems in weighted spaces. J. Differ. Equ. 250, 1235–1266 (2011)

Karachalios, N.I., Yannacopoulos, A.N.: Global existence and compact attractors for the discrete nonlinear Schrödinger equation. J. Differ. Equ. 217, 88–123 (2005)

Keener, J.P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556–572 (1987)

Nolen, J.: An invariance principle for random traveling waves in one dimension. SIAM J. Math. Anal. 43, 153–188 (2011)

Nolen, J., Xin, J.: A variational principle based study of KPP minimal front speeds in random shears. Nonlinearity 18, 1655–1675 (2005)

Nolen, J., Xin, J.: A variational principle for KPP front speeds in temporally random shear flows. Commun. Math. Phys. 269, 493–532 (2007)

Nolen, J., Xin, J.: Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 815–839 (2009)

Schmalfuss, B.: Backward cocycle and attractors of stochastic differential equations. In: Reitmann, V., Riedrich, T., Koksch, N. (eds.) International Seminar on Applied Mathematics-Nnonlinear Dynamics: Attractor Approximation and Global Behavior, pp. 185–192. Technische Universität, Dresden (1992)

Schmalfuss, B.: Lyapunov functions and non-trivial stationary solutions of stochastic differential equations. Dyn. Syst. 16(4), 303–317 (2001)

Shen, W.: Traveling waves in diffusive random media. J. Dynam. Differ. Equ. 16, 1011–1060 (2004)

Wang, B.: Dynamics of systems on infinite lattices. J. Differ. Equ. 221, 224–245 (2006)

Wang, B.: Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J. Differ. Equ. 253, 1544–1583 (2012)

Wang, B.: Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discret. Contin. Dyn. Syst. Ser. A 34, 269–300 (2014)

Wang, B.: Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations. Nonlinear Anal. 103, 9–25 (2014)

Wang, X., Li, S., Xu, D.: Random attractors for second-order stochastic lattice dynamical systems. Nonlinear Anal. 72, 483–494 (2010)

Wang, X., Lu, K., Wang, B.: Random attractors for delay parabolic equations with additive noise and deterministic non-autonomous forcing, Preprint, (2014)

Yan, W., Li, Y., Ji, S.: Random attractors for first order stochastic retarded lattice dynamical systems. J. Math. Phys. 51, 032702 (2010)

Zhang, Q., Zhao, H.: Stationary solutions of SPDEs and infinite horizon BDSDEs. J. Funct. Anal. 252, 171–219 (2007)

Zhang, Q., Zhao, H.: Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients. J. Differ. Equ. 248, 953–991 (2010)

Zhao, C., Zhou, S.: Attractors of retarded first order lattice systems. Nonlinearity 20, 1987–2006 (2007)

Zhao, C., Zhou, S.: Compact uniform attractors for dissipative lattice dynamical systems with delays. Discret. Contin. Dyn. Syst. 21, 643–663 (2008)

Zhou, S.: Attractors and approximations for lattice dynamical systems. J. Differ. Equ. 200, 342–368 (2004)

Zhou, S., Lu, W.: A random attractor for a stochastic second order lattice system with random coupled coefficients. J. Math. Anal. Appl. 395, 42–55 (2012)

Zinner, B.: Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Differ. Equ. 96, 1–27 (1992)