DOCSLIB.ORG

Comparison of Various Concepts of a Random Attractor: a Case Study

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Comparison of Various Concepts of a Random Attractor: a Case Study Arch. Math. 78 (2002) 233–240 Sonderdruck aus 0003-889X/02/030233-08 $ 3.10/0 Birkhäuser Verlag, Basel, 2002 Comparison of various concepts of a random attractor: A case study By MICHAEL SCHEUTZOW Abstract. Several non-equivalent definitions of an attractor of a random dynamical system have been proposed in the literature. We consider a rather simple special case: random dynamical systems with state space [0, ∞) which fix 0. We examine conditions under which the set {0} is an attractor for three different notions of an attractor. It turns out that even in this simple case the various concepts are quite different. The purpose of this note is to highlight these differences and thus provide a basis for discussion about the “correct” concept of a random attractor. 1. Introduction. While the concept of an attractor for deterministic dynamical systems is well-established, the same can not be said for random dynamical systems, which have gained considerable attention during the past decade (for a comprehensive survey see [1]). All definitions of a random attractor A (ω) known to the author agree in that they require that A (ω) be a random compact set which is invariant under the random dynamical system (below we will give precise definitions). The definitions disagree however with respect to the class of sets which are attracted as well as the precise meaning of “attracted”. In order to keep things reasonably simple we will always insist that an attractor attracts all (deterministic) bounded subsets of the (metric) state space. Furthermore we will only discuss random dynamical systems in continuous time. Out of the three definitions we give below the notion of a forward attractor is closest to that of an attractor for a deterministic dynamical system. It is however believed to be the least appropriate one for random dynamical systems. The concept of a pullback attractor (also called strong attractor or just attractor) was proposed independently by H. Crauel and F. Flandoli [3] and B. Schmalfuß [11]. Weak attractors were recently introduced by G. Ochs [10]. In his paper he highlights differences between weak and pullback attractors e.g. concerning invariance properties under random transformations. It is not our aim to point out such different properties but rather to illustrate the different concepts within a particular class of one-dimensional random dynamical systems. For the related question of bifurcations of one-dimensional stochastic differential equations, see [4] and [12]. Before we provide precise definitions we recall the notion of a random dynamical system taking values in a space E. In the second section we will then consider the particular case E =[0, ∞). Mathematics Subject Classification (2000): Primary 37H05, 60H10; Secondary 60G44, 60J55. 234 M. SCHEUTZOW ARCH. MATH. Throughout the paper we assume that (E,d) is a separable complete metric space. Whenever necessary we will equip E with the Borel-σ-algebra E (i.e. the smallest σ-algebra on E which contains all open subsets). D e f i n i t i o n 1.1. A random dynamical system (RDS) with state space E is a pair (ϑ, φ) consisting of the following two objects: 1. A metric dynamical system (MDS) ϑ ≡ (Ω, F, P, {ϑ(t), t ∈ R}), i.e. a probability space (Ω, F, P) with a family of measure preserving transformations {ϑ(t) : Ω → Ω, t ∈ R} such that (a) ϑ(0) = idΩ,ϑ(t) ◦ ϑ(s) = ϑ(t + s) for all t, s ∈ R; (b) the map (t,ω) → ϑ(t)ω is measurable and ϑ(t)P = P for all t ∈ R. 2. A (perfect) cocycle φ over ϑ of continuous mappings of X i.e. a measurable map φ : R+ × Ω × E → E,(t,ω,x) → φ(t,ω,x) such that (a) the mapping φ(·,ω,·) : (t, x) → φ(t,ω,x) is continuous for all ω ∈ Ω and (b) it satisfies the cocycle property: φ(0,ω,·) = idE,φ(t + s,ω,·) = φ(t,ϑ(s)ω, φ(s,ω,·)) for all t, s м 0 and ω ∈ Ω. D e f i n i t i o n 1.2. Let (ϑ, φ) be an RDS and ω → A (ω) satisfy (i) A (ω) is a compact subset of E for every ω ∈ Ω (ii) ω → d(x, A (ω)) is measurable for each x ∈ E (iii) φ(t,ω,A (ω)) = A (ϑ(t)ω) for all ω ∈ Ω, t м 0. Then A is called an invariant random compact set. D e f i n i t i o n 1.3. Let A be an invariant random compact set of the RDS (ϑ, φ). A is called a forward attractor if for each bounded set B ֤ E .1 lim sup d(φ(t,ω,x), A (ϑ(t)ω)) = 0 a.s. →∞ t x∈B A is called a pullback attractor if for each bounded set B ֤ E .2 lim sup d(φ(t,ϑ(−t)ω, x), A (ω)) = 0 a.s. →∞ t x∈B A is called a weak attractor if for each bounded set B ֤ E .3 lim sup d(φ(t,ω,x), A (ϑ(t)ω)) = 0 in probability. →∞ t x∈B Remark 1.4.Obviouslyallthreeconceptscoincideinthedeterministiccase(i.e. Ω is a singleton). In general each forward attractor and each pullback attractor is a weak attractor (the latter implication holds since ϑ(t) preserves P). No other implication holds as we will see in the next section. It follows from [10], Theorem 1 and Corollary 3.2, that an attractor (in whatever sense) – if it exists – is unique up to a set of measure 0. Some authors only require that an attractor attracts all compact subsets instead of all bounded ones or even just all singletons, e.g. [10]. Since we will only consider the special case E =[0, ∞) below, these distinctions will be irrelevant for us. Vol. 78, 2002 Random attractors 235 2. A case study. Let E =[0, ∞) be equipped with the euclidean metric. We will restrict our attention to RDSs on E which are generated by stochastic differential equations (SDEs) of the form (1) dX(t) = b(X(t))dt + σ(X(t)) ◦ dW(t) = ( ( )) + 1 σ( ( ))σ ( ( )) + σ( ( )) ( ), (2) b X t dt 2 X t X t X t dW t where W(t), t ∈ R is a two-sided standard (real-valued) Brownian motion. The symbol “◦” denotes the Stratonovich stochastic integral and the last equality sign holds due to the well- known transformation formula between the Itô and the Stratonovich integral ([7], p. 296). We will assume throughout that the following conditions on the coefficients b : E → R and σ : E → R are satisfied. • b(0) = σ(0) = 0 • b satisfies a Lipschitz condition on each compact subset of E and grows at most linearly • σ ∈ C2 with bounded first and second derivatives • σ(x)>0 for all x > 0. In order to arrive at an RDS, we choose specifically Ω = C(R, R) – the space of continuous functions from R to R, F the σ-field on Ω generated by the evaluations (or the topology of uniform convergence on compact subsets), P two-sided Wiener measure on (Ω, F) and define ϑ(t) : Ω → Ω by (ϑ(t)(ω)) (s) := ω(t + s) − ω(t), s, t ∈ R. Then ϑ ≡{Ω, F, P,ϑ(t), t ∈ R} is a metric dynamical system and W(t,ω)= ω(t), t ∈ R is a two-sided Brownian motion. It is shown in [2] and [1] that one can find a cocycle φ of homeomorphisms of E on the MDS ϑ such that for each x ∈ E, φ(t,ω,x), t м 0 solves (1) with initial condition X(0) = x. More generally, for every s ∈ R, φ(t − s,ϑ(s)(ω), x), t м s solves (1) with initial condition X(s) = x. Our assumptions on b and σ above imply (after changing φ on a set of measure 0 if necessary) that φ(t,ω,0) = 0 for all t,ω. We can extend φ to a map φ : R × Ω × E → E by defining (3) φ(−t,ω,x) := φ−1(t,ϑ(−t)(ω), .)(x), t м 0. Then φ satisfies the second part of Definition 1.1 even for all s, t ∈ R,i.e.φ is a cocycle of homeomorphisms with index set R. Note that this implies that φ is order preserving i.e. x > y м 0 implies φ(t,ω,x)>φ(t,ω,y) for all t ∈ R,ω∈ Ω. We will be interested in explicit necessary and/or sufficient conditions on the functions b and σ which guarantee that A (ω) ={0} is a (weak, pullback or forward) attractor of (ϑ, φ). Equation (1) does not only generate an RDS, it also generates a diffusion process on E. The probability law of a diffusion on E is uniquely characterized by its scale function p :[0, ∞] → R ∪{±∞}and its speed measure m(dx) on (0, ∞) defined as x v 1 2b(u) (4) p(x) = exp − du dv σ(v) σ 2(u) 1 1 x 1 2b(u) (5) m(dx) = exp du dx. σ(x) σ 2(u) 1 236 M. SCHEUTZOW ARCH. MATH. The relevance of the function p is, that p(X(t)), t м 0 becomes a continuous local mar- tingale. The finiteness or not of p(0), p(∞), m(0, 1] and m[1, ∞) is closely linked with the behavior of the diffusion at or close to the boundary {0, ∞} of E, see e.g.
Load more

Recommended publications
  • Forward Attraction of Pullback Attractors and Synchronizing Behavior of Gradient-Like Systems with Nonautonomous Perturbations X
    DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2019011 DYNAMICAL SYSTEMS SERIES B Volume 24, Number 3, March 2019 pp. 1175{1197 FORWARD ATTRACTION OF PULLBACK ATTRACTORS AND SYNCHRONIZING BEHAVIOR OF GRADIENT-LIKE SYSTEMS WITH NONAUTONOMOUS PERTURBATIONSy Xuewei Ju Department of Mathematics, School of Science Civil Aviation University of China Tianjin 300300, China Desheng Li∗ School of Mathematics, Tianjin University Tianjin 300072, China Jinqiao Duan Department of Applied Mathematics Illinois Institute of Technology Chicago, IL 60616, USA Abstract. We consider the nonautonomous perturbation xt + Ax = f(x) + "h(t) of a gradient-like system xt + Ax = f(x) in a Banach space X, where A is a sectorial operator with compact resolvent. Assume the non-perturbed system xt + Ax = f(x) has an attractor A . Then it can be shown that the perturbed one has a pullback attractor A" near A . If all the equilibria of the non-perturbed system in A are hyperbolic, we also infer from [4,6] that A" inherits the natural Morse structure of A . In this present work, we introduce the notion of nonautonomous equilibria and give a more precise description on the Morse structure of A" and the asymptotically synchronizing behavior of the perturbed system. Based on the above result we further prove that the sections of A" depend on time symbol continuously in the sense of Hausdorff distance. Consequently, one concludes that A" is a forward attractor of the perturbed nonautonomous system. It will also be shown that the perturbed system exhibits completely a global forward synchronizing behavior with the external force. 1. Introduction.
    [Show full text]
  • Attractors for Ergodic and Monotone Random Dynamical Systems
    Attractors for ergodic and monotone random dynamical systems Michael Scheutzow, Technische Universit¨at Berlin May 26, 2005 In this talk, we relate ergodicity, monotonicity and attractors of a random dynamical system (rds). The main result is that under suitable conditions, ergodicity and monotonicity together imply the existence of a weak random attractor which consists of a single random point only. We also show that ergodicity alone does not suffice to guarantee the existence of a weak attractor { not even when the one-point motion is a diffusion on Rd for d ≥ 2. The results can be applied to a large class of systems, eg to certain SPDEs. The talk is largely based on joint work with Igor Chueshov (Kharkov), see [2]. Our motivation for studying this problem is to provide a rather gen- eral sufficient condition for a random attractor. Indeed several authors have proved the existence of random attractors for particular systems using ad hoc methods and in many cases the systems were in fact monotone and ergodic. When an rds is monotone, then this property is usually very easy to prove. Ergodicity is not always easy to prove { but in most cases still considerably easier than to prove the existence of an attractor. The result below provides a stronger conclusion than most other results on attractors in that we show, that the attractor is trivial (i.e. consists of a single point). On the other hand, due to the generality of the set-up, we can only show that the attrac- tor attracts all deterministic compact sets in probability while some authors prove almost sure attraction for a larger class of sets.
    [Show full text]
  • Pullback Attractors for Bi-Spatial Continuous Random Dynamical Systems and Application to Stochastic Fractional Power Dissipative Equation on an Unbounded Domain
    DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018326 DYNAMICAL SYSTEMS SERIES B Volume 24, Number 7, July 2019 pp. 3395{3438 PULLBACK ATTRACTORS FOR BI-SPATIAL CONTINUOUS RANDOM DYNAMICAL SYSTEMS AND APPLICATION TO STOCHASTIC FRACTIONAL POWER DISSIPATIVE EQUATION ON AN UNBOUNDED DOMAIN Wenqiang Zhao School of Mathematics and Statistics Chongqing Technology and Business University Chongqing 400067, China (Communicated by Bj¨ornSchmalfuß) Abstract. In this article, a notion of bi-spatial continuous random dynami- cal system is introduced between two completely separable metric spaces. It is show that roughly speaking, if such a random dynamical system is asymptoti- cally compact and random absorbing in the initial space, then it admits a bi- spatial pullback attractor which is measurable in two spaces. The measurability of pullback attractor in the regular spaces is completely solved theoretically. As applications, we study the dynamical behaviour of solutions of the non- N autonomous stochastic fractional power dissipative equation on R with addi- tive white noise and a polynomial-like growth nonlinearity of order p; p ≥ 2. We 2 N s N p N prove that this equation generates a bi-spatial (L (R );H (R ) \ L (R ))- continuous random dynamical system, and the random dynamics for this sys- tem is captured by a bi-spatial pullback attractor which is compact and at- s N p N s N tracting in H (R ) \ L (R ), where H (R ) is a fractional Sobolev space with s 2 (0; 1). Especially, the measurability of pullback attractor is individu- s N p N ally derived by proving the the continuity of solutions in H (R ) and L (R ) with respect to the sample.
    [Show full text]
  • Connectedness of Random Set Attractors,Bymichael Scheutzow and Isabell Vorkastner, Communicated on April 20, 2018.1
    Rend. Lincei Mat. Appl. 29 (2018), 607–617 DOI 10.4171/RLM/824 Probability Theory — Connectedness of random set attractors,byMichael Scheutzow and Isabell Vorkastner, communicated on April 20, 2018.1 Abstract. — We examine the question whether random set attractors for continuous-time ran- dom dynamical systems on a connected state space are connected. In the deterministic case, these attractors are known to be connected. In the probabilistic setup, however, connectedness has only been shown under stronger connectedness assumptions on the state space. Under a weak continuity condition on the random dynamical system we prove connectedness of the pullback attractor on a connected space. Additionally, we provide an example of a weak random set attractor of a random dynamical system with even more restrictive continuity assumptions on an even path-connected space which even attracts all bounded sets and which is not connected. On the way to proving con- nectedness of a pullback attractor we prove a lemma which may be of independent interest and which holds without the assumption that the state space is connected. It states that even though pullback convergence to the attractor allows for exceptional nullsets which may depend on the com- pact set, these nullsets can be chosen independently of the compact set (which is clear for s-compact spaces but not at all clear for spaces which are not s-compact). Key words: Random dynamical system, pullback attractor, weak attractor, pullback continuity, measurable selection Mathematics Subject Classification: 37H99, 37B25, 37C70, 28B20 1. Introduction While attractors for (deterministic) dynamical systems have been studied for a long time, attractors for random dynamical systems were only introduced and studied in the nineties of the last century.
    [Show full text]
  • Probabilistic Continuity of a Pullback Random Attractor in Time-Sample
    DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2020028 DYNAMICAL SYSTEMS SERIES B Volume 25, Number 7, July 2020 pp. 2699{2722 PROBABILISTIC CONTINUITY OF A PULLBACK RANDOM ATTRACTOR IN TIME-SAMPLE Shulin Wang Faculty of Eduction Southwest University Chongqing 400715, China Yangrong Li∗ School of Mathematics and Statistics Southwest University Chongqing 400715, China (Communicated by Bj¨ornSchmalfuß) Abstract. Given a time-sample dependent attractor of a random dynamical system, we study its lower semi-continuity in probability along the time axis, and the criteria are established by using the local-sample asymptotically com- pactness for a triple-continuous system. The abstract results are applied to the non-autonomous stochastic p-Laplace equation on an unbounded domain with weakly dissipative nonlinearity. Without any additional hypotheses, we prove that the pullback random attractor is probabilistically continuous in both time and sample parameters. 1. Introduction. A core concept in the theory of non-autonomous random dy- namical systems is the so-called pullback random attractor (PRA), which seems to be first involved in Crauel et al. [8] and systemically studied by Wang [39]. A PRA is formulated by a bi-parametric set A = fA(τ; !)g, where τ 2 R is the current time and ! is the sample in a measurable dynamical system (Ω; F; P; θt). Its properties for a fixed time-fiber had been researched, see [10, 11, 12, 29,42, 45]. Recently, Cui, Kloeden and Wu[14] have studied the upper semi-continuity of a PRA in time-parameter: lim dist(A(τ; !); A(τ0;!)) = 0; 8τ0 2 R;! 2 Ω: (1) τ!τ0 In this paper, we consider a more challenging problem on the lower semi-continuity: lim P f! 2 Ω : dist(A(τ0;!); A(τ; !)) ≥ δg = 0; 8δ > 0; τ0 2 R: (2) τ!τ0 In general, it is impossible to obtain the lower semi-continuity even if the upper semi-continuity holds true, many examples can be found in [19, 32, 33].
    [Show full text]
  • The Relationship Between Pullback, Forward and Global Attractors of Nonautonomous Dynamical Systems
    Nonlinear Dynamics and Systems Theory, 2(2) (2002) 125–144 The Relationship between Pullback, Forward and Global Attractors of Nonautonomous Dynamical Systems D.N. Cheban1, P.E. Kloeden2 and B. Schmalfuß3† 1Departement of Mathematics and Informatics, State University of Moldova, A. Mateevich Street 60, MD–2009 Chi¸sin˘au, Moldova 2Fachbereich Mathematik, Johann Wolfgang Goethe Universit¨at, D–60054 Frankfurt am Main, Germany 3Department of Sciences, University of Applied Sciences, D–06217 Merseburg, Germany Received: July 18, 2001; Revised: June 12, 2002 Abstract: Various types of attractors are considered and compared for non- autonomous dynamical systems involving a cocycle state space mapping that is driven by an autonomous dynamical system on a compact metric space. In particular, conditions are given for a uniform pullback attractor of the cocycle mapping to form a global attractor of the associated autonomous skew-product semi-dynamical system. The results are illustrated by several examples that are generated by differential equations on a Banach space with a uniformly dissipative structure induced by a monotone operator. Keywords: Nonautonomous dynamical system; skew-product flow; pullback attrac- tor; global attractor; asymptotical stability; nonautonomous Navier-Stokes equation. Mathematics Subject Classification (2000): 34D20, 34D40, 34D45, 58F10, 58F12, 58F39, 35B35, 35B40. 1 Introduction Nonautonomous dynamical systems can often be formulated in terms of a cocycle map- ping for the dynamics in the state space that is driven by an autonomous dynamical system in what is called a parameter or base space. Traditionally the driving system †This work was partially supported by the DFG Forschungsschwerpunktsprogramm “Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme” and “Interagierende stochastische Systeme von hoher Komplexit¨at”.
    [Show full text]
  • Random Pullback Attractor of a Non-Autonomous Local Modified Stochastic Swift-Hohenberg with Multiplicative Noise
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 February 2020 doi:10.20944/preprints202002.0350.v1 Article Random pullback attractor of a non-autonomous local modified stochastic Swift-Hohenberg with multiplicative noise Yongjun Li 1*, Tinggang Zhao1 and Hongqing Wu1 1 School of Mathematics, Lanzhou City University, Lanzhou, 730070, P.R. China; [email protected] (Y. Li); [email protected](T.Zhao); [email protected](H.Wu), * Correspondence: [email protected] (Y. Li) Abstract: In this paper, we study the existence of the random -pullback attractor of a non-autonomous local modified stochastic Swift-Hohenberg equation with multiplicative noise in 2 stratonovich sense. It is shown that a random -pullback attractor exists in H0 (D) when its external force has exponential growth. Due to the stochastic term, the estimate are delicate, we overcome this difficulty by using the Ornstein-Uhlenbeck(O-U) transformation and its properties. Keywords: Swift-Hohenberg equation; Random -pullback attractor; Non-autonomous random dynamical system 1. Introduction The Swift-Hohenberg(S-H) type equations arise in the study of convective hydrodynamical, plasma confinement in toroidal and viscous film flow, was introduced by authors in [1]. After that, Doelman and Standstede [2] proposed the following modified Swift-Hohenberg equation for a pattern formation system near the onset to instability 2 2 3 ut + u +2 u + au + b | u | + u = 0, (1.1) where a and b are arbitrary constants. We can see from the above equation that there exist two operators and 2 , these two operators 1 2 have some symmetry, for any u H0 () D , the inner product (,)(,)u u = − u u =‖u‖ , for any 2 2 2 2 u H0 () D , (u , u ) = ( u , u )=‖u‖ , is antisymmetry and is symmetry.
    [Show full text]
  • The Dynamics of Random Difference Equations Is Remodeled by Closed Relations∗
    SIAM J. MATH. ANAL. c 2014 Society for Industrial and Applied Mathematics Vol. 46, No. 1, pp. 459–483 THE DYNAMICS OF RANDOM DIFFERENCE EQUATIONS IS REMODELED BY CLOSED RELATIONS∗ G. MANJUNATH† AND H. JAEGER† Abstract. We provide substantial simplifications in the understanding of qualitative dynamics of random difference equations (RDEs) by constructing closed relations using their typical instanti- ations. We consider the RDE given by xn+1 = g(ξn(ω),xn), where g : U × X → X is a uniformly continuous map, {ξn} is a U-valued stationary input, and X a compact metric space. We represent each nonautonomous difference equation (NDE) {gn,ω (·):=g(ξn(ω), ·):n ∈ Z} obtained by any realization of the input by some closed relation and prove a number of useful results. For all typical realizations of the input process, we show (1) any solution of the NDE can be obtained as an orbit (trajectory) of the corresponding closed relation; (2) every attractor A (an autonomous subset of X) of the closed relation contains a positively invariant uniform attractor and a pullback attractor of the NDE; (3) every (entire) solution of the NDE converges to an A in (2) or else stays in its dual repeller; (4) the closed relations are all identical whenever the input process is ergodic. Uniformity is a highly relevant condition for a nonautonomous attractor to be of practical relevance. Statement (3) yields a “Conley-like” decomposition theorem in terms of autonomous subsets alone and leads to a remarkable simplification in understanding the asymptotic dynamics of each such NDE. Key words.
    [Show full text]
  • Existence of D−Pullback Attractor for an Infinite Dimensional Dynamical System
    DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2021036 DYNAMICAL SYSTEMS SERIES B EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN INFINITE DIMENSIONAL DYNAMICAL SYSTEM Mustapha Yebdri Laboratory of dynamical systems Department of Mathematics,Faculty of sciences University of Tlemcen Tlemcen, BP.119, 13000 Algeria (Communicated by Jos´eA. Langa) Abstract. At the very beginning of the theory of finite dynamical systems, it was discovered that some relatively simple systems, even of ordinary differen- tial equations, can generate very complicated (chaotic) behaviors. Furthermore these systems are extremely sensitive to perturbations, in the sense that trajec- tories with close but different initial data may diverge exponentially. Very of- ten, the trajectories of such chaotic systems are localized, up to some transient process, in some subset of the phase space, the so-called strange attractors. Such subset have a very complicated geometric structure. They accumulate the nontrivial dynamics of the system. For a distributed system, whose time evolution is usually governed by par- tial differential equations (PDEs), the phase space X is (a subset of) an infinite dimensional function space. We will thus speak of infinite dimensional dynam- ical systems. Since the global existence and uniqueness of solutions has been proven for a large class of PDEs arising from different domains as Mechanics and Physics, it is therefore natural to investigate whether the features, in par- ticular the notion of attractor, obtained for dynamical systems generated by systems of ODEs generalizes to systems of PDEs. In this paper we give a positive aftermath by proving the existence of pull- back D-attractor. The key point is to find a bounded family of pullback D- absorbing sets then we apply the decomposition techniques and a method used in previous works to verify the pullback w-D-limit compactness.
    [Show full text]
  • Minimal Random Attractors
    Minimal random attractors Hans Crauel∗ Michael Scheutzow† It is well-known that random attractors of a random dynamical system are generally not unique. We show that for general pullback attractors and weak attractors, there is always a minimal (in the sense of smallest) random at- tractor which attracts a given family of (possibly random) sets. We provide an example which shows that this property need not hold for forward attrac- tors. We point out that our concept of a random attractor is very general: The family of sets which are attracted is allowed to be completely arbitrary. 2010 Mathematics Subject Classification Primary 37H99 Secondary 34D55 35B51 37B25 37C70 37L99 60H10 60H15 60H25 Keywords. Random attractor; pullback attractor; weak attractor; forward attractor; Omega limit set; compact random set; closed random set; closed random hull 1 Introduction For deterministic dynamical systems on metric spaces the notion of an attractor is well established. The most common notion, mainly used for partial differential equations (PDEs) on suitable Hilbert or Banach spaces, is that of a global set attractor. It is char- acterized by being a compact set, being strictly invariant, and attracting every compact, or even every bounded set. Uniqueness of the global set attractor is immediate. Another very common notion is that of a (global) point attractor, which is often used for systems on locally compact spaces (which are often Euclidean spaces or finite- dimensional manifolds). A global point attractor is again compact and strictly invariant, arXiv:1712.08692v1 [math.DS] 23 Dec 2017 but it is only assumed to attract every point (or, equivalently, every finite set).
    [Show full text]