Minimal random attractors

Abstract

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 attractor which attracts a given family of (possibly random) sets. We provide an example which shows that this property need not hold for forward attractors. 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.

Random dynamics of fractional nonclassical diffusion equations driven by colored noise

“…We remark that for stochastic PDEs with standard Laplacian, random attractors have been investigated in [8,12,13,14,15,18,20,21,23,31,35,36,47,48,53,57,58] for the autonomous case, and in [1,9,10,24,25,19,30,33,34,37,49,62,63] for the non-autonomous case. The reader is referred to [26,28,38,55] for attractors of random systems with standard Laplacian driven by colored noise or approximations of white noise.…”

Random perturbations of an eco-epidemiological model

“…The attractor need not be unique for a general family R. However, as soon as R contains every compact deterministic set, if a random attractor for R exists then it is unique (cf. [13]). Notice this is the case if R ∈ {D(X), T (X)}.…”

“…Random attractors are a central concept in the analysis of random models. Since their introduction there are several improvements regarding the existence and properties of such attractors, but there are questions that are still open in this theory, namely on bifurcations and their random attractors, on system with random pullback attractors which are not pathwise forward attracting and vice-versa, on the structure of random attactors as a finer description of their dynamics for instance via Morse decomposition, particularly in infinite dimensional phase space, and also concerning efficient numerical computation of pullback convergence, among others; see [9,10,11,12,13,18]. The main strategy adopted to ensure the existence of a random attractor for a given family of random sets is to find a compact absorbing set.…”

Minimal forward random point attractors need not exist

“…Here we provide an example which shows that the answer is negative. The example is an appropriate modification of the example given in [1].…”