Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise☆
Introduction
It is well known that stochastic lattice differential equations arise naturally in a wide variety of applications where the spatial structure has a discrete character and uncertainties or random influences, called noises, are taken into account. The lattice differential equations have been used to model systems such as cellular neural networks with applications to image processing, pattern recognition, and brain science (see [13] and references therein). They are also used to model the propagation of pulses in myelinated axons where the membrane is excitable only at spatially discrete sites, in which case represents the potential at the th active site (see e.g., [6], [25]). Lattice differential equations can also be found in chemical reaction theory (see e.g., [17], [24]). In the absence of noise, many works have been done on various aspects of solutions to deterministic lattice dynamical systems. We refer the readers to [3], [26] and references therein for traveling waves, and [12], [30] and references therein for the chaotic properties of solutions.
In this paper we will investigate the long term behavior for the following stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term and a different multiplicative white noise at each node: where denotes the integer set, , , and are positive constants, is a smooth function satisfying proper dissipative conditions, and ’s are mutually independent Brownian motions. Here denotes the Stratonovich sense of the stochastic term.
The study of global random attractors was initiated by Ruelle [27]. The fundamental theory of global random attractors for stochastic partial differential equations was developed by Crauel and Flandoli [14], Flandoli and Schmalfuss [18], Imkeller and Schmalfuss [21], Schmalfuss [28] amongst others. Due to the unbounded fluctuations in the systems caused by the white noise, the concept of global pullback/random attractor was introduced to capture the essential dynamics with possibly extremely wide fluctuations. This is significantly different from the deterministic case.
The existence of a global attractor for the deterministic counterpart of (1.1) was established in [5]. 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. [4], Caraballo et al. [10], [11], Caraballo and Lu [9], Han [19], Han et al. [20], 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 Eq. (1.1) is different at each node. This can be the result of an environmental effect on the whole domain of the system, either on each equation of the lattice (additive noise case) [4] or in some parameters of the model (multiplicative noise case). To be more precise, and focus on the multiplicative case, let us recall that in some previous lattice models analyzed in the existing literature, the term which is responsible for the dissipative character of the problem is sometimes split into two terms, one of which is linear. For that reason, the deterministic counterpart of our model (1.1) is given as (see, e.g. Caraballo and Lu [9]) Assuming now that the environmental effect produces a perturbation in the parameter in such a way that it becomes at each node , then our stochastically perturbed lattice becomes which is precisely the model we are interested in, if we denote .
In the present paper we will prove the existence of a global random attractor for the infinite dimensional random dynamical system generated by the stochastic lattice differential equation (1.1). An interesting feature of this structure is that, even though the spatial domain is unbounded and the solution operator is not smooth or compact, unlike parabolic type of partial differential equations on bounded domains, bounded sets of initial data converge in the pullback sense, under the forward flow to a random compact invariant set. It is worth mentioning again that the noise involved in the system is multiplicative, and more importantly, different at each node. We put emphasis again on this comment because, to the best of our knowledge, such systems have not been considered in the literature although they are fully justified by physical intuitions. More precisely, all the papers published on this topic to date consider at most a finite sum of noise in each node which means that the noise term is essentially the same at all the nodes.
In previous studies in the literature, the stochastic systems are first rewritten as a stochastic differential equation in the Hilbert space , then a suitable change of variable, involving usually an Ornstein–Uhlenbeck process, is performed in order to transform the stochastic equation into a random differential equation in . This has subsequently become a standard way of formalization (see e.g. [9], [11], [20]). 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.
Unlike the previously described technique, we will perform a change of variable first to transform the stochastic lattice system (1.1) into a random lattice system. We then prove that the random dynamical system generated by the resulting random lattice system possesses a global random attractor. The existence of global random attractors for the original stochastic lattice system can be obtained once it can be shown that the random dynamical system generated by the transformed equation is conjugated with the original one in the same space of sequences .
After the transformation, we need to formulate the random lattice system as an abstract evolution equation in a Gelfand triple of Hilbert spaces formed by sequences. Instead of carrying out our analysis working only for our particular random lattice system, we will first develop an abstract theory for the existence of weak solutions1 to general random differential equations defined in a Gelfand triple of Hilbert spaces. Then we will apply it to our lattice model as a particular example. For this reason the goal of this paper is two-fold:
- 1.
proving a general theorem on the existence and uniqueness of weak solutions for random differential equations in Hilbert spaces;
- 2.
proving, as a special application, that our stochastic lattice model (1.1) generates, after a suitable change of variable, a random dynamical system which possesses a global random attractor.
The rest of the paper is organized as follows. In Section 2, we introduce basic concepts concerning random dynamical systems and global random attractors. In Section 3, we perform a transformation which allows us to rewrite our lattice system as a random lattice system without white noise, which can be eventually written as an evolution equation in some appropriate Hilbert spaces. Section 4 is devoted to establishing the existence and uniqueness of weak solutions for general abstract random evolution equations, which will be used later to prove that (1.1) generates an infinite dimensional random dynamical system. The existence of the global random attractor for (1.1) is finally established in Section 5. Some closing remarks will be given in Section 6.
Section snippets
Preliminaries on random dynamical systems
We recall in this section some of the basic concepts related to random dynamical systems and the concept of random attractors (see [2], [7], [18] for more details).
Let be a separable Banach space and a probability space. Definition 2.1 is called a metric dynamical system, if is -measurable, is the identity on , for all , for all .
Consider the probability space , where endowed with the compact
Mathematical preparation
Our goal is to study the global random attractor for the random dynamical system generated by (1.1). To this end we will first transform the stochastic lattice equation (1.1) containing white noise terms into a random lattice equation without white noise terms but with random coefficients, which can be written as a random evolution equation eventually. We will then investigate the random attractor for the resulting random evolution equation.
A finite version of system (1.1) as follows
Weak solutions to general random evolution equations
In this section, we will state and prove an existence theorem for weak solutions to general random evolution equations with particular type of operators and study the random dynamical systems generated by the weak solutions.
Let be a separable Hilbert space equipped with the inner product and the norm . Let be a dense subspace of with the inner product and the norm , and assume that is given a topological vector space structure for which the inclusion map is
Existence of global random attractors for the stochastic lattice dynamical system
We will now apply the general results proved in Section 4 to our special lattice equation (3.3). To this end, set with inner product Denote by the element in with value 1 at position and 0 for all other components. Let be a sequence of positive numbers. In particular, we assume that and in addition that for some positive , which ensures that
Closing remarks
This work is motivated by realizing the physical limitation of considering exactly the same multiplicative noise at each node in a lattice system under random influences. In particular, when the randomness comes from an environmental noise that affects the whole system, but differently at each node, it is more realistic to consider a different noise (but with similar structure) at different node. Driven by this motivation, we studied in this work a stochastic lattice dynamical system with
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This work has been partially supported by FEDER and Ministerio de Economía y Competitividad (Spain) under grants MTM2011-22411 and MTM2012-31698, and by Junta de Andalucía under Proyecto de Excelencia P12-FQM-1492.