Random attractors for stochastic lattice dynamical systems in weighted spaces

Han, Xiaoying; Shen, Wenxian; Zhou, Sizhong

doi:10.1016/j.jde.2010.10.018

Cited by 164 publications

(82 citation statements)

References 35 publications

(61 reference statements)

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“…Note that in [19], [21] and any other existing work of stochastic lattice systems with single or a finite sum of noise at all nodes, the system can be reformulated as a stochastic equation in the (regular or weighted) space of infinite sequences, before the change of variable process transforming the stochastic equation into a random equation. However, due to the infinite number of noises in this problem, we have to perform the change of variables to (1.1) before formulating it as an evolution equation in appropriate spaces of infinite sequences.…”

Section: Mathematical Preparationmentioning

confidence: 99%

“…For stochastic lattice dynamical systems with additive or multiplicative noise, the existence of global random attractors has been intensively analyzed in the recent literature (see e.g., Bates et al [16], Caraballo et al [17,18], Caraballo and Lu [19], Han [20], Han et al [21], amongst others). We emphasize that in the studies of stochastic lattice systems with multiplicative noise up to date, only a finite number of Wiener process is considered in each equation, being the same in all the equations, while the multiplicative noise considered here in Equation (1.1) is different at each node.…”

Section: Introductionmentioning

confidence: 99%

“…This has subsequently become a standard way of formalization (see e.g. [19,18,21]). However, due to the appearance of the infinitely many noise terms, this scheme cannot be applied to handle our problem and hence a new methodology is ought to be developed.…”

Section: Introductionmentioning

confidence: 99%

“…In this way we avoid the calculation of uniform estimates on the tails of the solutions, as it has been done in all previous published works on this topic in order to prove either asymptotic compactness [19] or asymptotic nullness [21].…”

Section: Introductionmentioning

confidence: 99%

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Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

Caraballo

Han

Schmalfuß³

et al. 2016

Nonlinear Analysis

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In this paper we investigate the long term behavior of a stochastic lattice dynamical system with a diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and a different multiplicative white noise at each node. We prove that this stochastic lattice equation generates a random dynamical system that possesses a global random attractor. In particular, we first establish an existence theorem for weak solutions to general random evolution equations, which is later applied to the specific stochastic lattice system to show that it has weak solutions and the solutions generate a random dynamical system. We then prove the existence of a random attractor of the underlying random dynamical system by constructing a random compact absorbing set and using an embedding theorem. The major novelty of this work is that we consider a different multiplicative white noise term at each different node, which significantly improves the previous results in the literature where the same multiplicative noise was considered at all the nodes. As a consequence, the techniques used in the existing literature are not applicable here and a new methodology has to be developed to study such systems.

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Section: Mathematical Preparationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

Caraballo

Han

Schmalfuß³

et al. 2016

Nonlinear Analysis

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“…Usually, the models under consideration are obtained by a spatial discretization of a parabolic or a hyperbolic equation (see e.g. [1], [2], [4], [5], [8], [11] [12], [15], [16], [19], [20], [22], [23], [26], [28], [29]). …”

Section: Introductionmentioning

confidence: 99%

Attractors for non-autonomous retarded lattice dynamical systems

Caraballo

Morillas

Valero

2015

Nonautonomous Dynamical Systems

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Abstract:In this paper we study a non-autonomous lattice dynamical system with delay. Under rather general growth and dissipative conditions on the nonlinear term, we de ne a non-autonomous dynamical system and prove the existence of a pullback attractor for such system as well. Both multivalued and single-valued cases are considered.Keywords: lattice dynamical systems, non-autonomous systems, di erential equations with delay, set-valued dynamical systems, pullback attractor MSC: 34K05, 34K31, 35B40, 35B41, 35K55, 35K40, 37L30, 58C06

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Lattice Dynamical Systems in the Biological Sciences

Han

Kloeden

2019

The IMA Volumes in Mathematics and Its Applications

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Random attractors for stochastic lattice dynamical systems in weighted spaces

Cited by 164 publications

References 35 publications

Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

Attractors for non-autonomous retarded lattice dynamical systems

Lattice Dynamical Systems in the Biological Sciences

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