# Pullback attractor

In mathematics, the attractor of a random dynamical system may be loosely thought of as a set to which the system evolves after a long enough time. The basic idea is the same as for a deterministic dynamical system, but requires careful treatment because random dynamical systems are necessarily non-autonomous. This requires one to consider the notion of a pullback attractor or attractor in the pullback sense.

## Set-up and motivation

$d\left(\varphi (t_{n},\omega )x_{0},a\right)\to 0$ as $n\to \infty$ .

This is not too far from a working definition. However, we have not yet considered the effect of the noise $\omega$ , which makes the system non-autonomous (i.e. it depends explicitly on time). For technical reasons, it becomes necessary to do the following: instead of looking $t$ seconds into the "future", and considering the limit as $t\to +\infty$ , one "rewinds" the noise $t$ seconds into the "past", and evolves the system through $t$ seconds using the same initial condition. That is, one is interested in the pullback limit

$\lim _{t\to +\infty }\varphi (t,\vartheta _{-t}\omega )$ .

So, for example, in the pullback sense, the omega-limit set for a (possibly random) set $B(\omega )\subseteq X$ is the random set

$\Omega _{B}(\omega ):=\left\{x\in X\left|\exists t_{n}\to +\infty ,\exists b_{n}\in B(\vartheta _{-t_{n}}\omega )\mathrm {\,s.t.\,} \varphi (t_{n},\vartheta _{-t_{n}}\omega )b_{n}\to x\mathrm {\,as\,} n\to \infty \right.\right\}.$ Equivalently, this may be written as

$\Omega _{B}(\omega )=\bigcap _{t\geq 0}{\overline {\bigcup _{s\geq t}\varphi (s,\vartheta _{-s}\omega )B(\vartheta _{-s}\omega )}}.$ Importantly, in the case of a deterministic dynamical system (one without noise), the pullback limit coincides with the deterministic forward limit, so it is meaningful to compare deterministic and random omega-limit sets, attractors, and so forth.

## Definition

The pullback attractor (or random global attractor) ${\mathcal {A}}(\omega )$ for a random dynamical system is a $\mathbb {P}$ -almost surely unique random set such that

$\lim _{t\to +\infty }\mathrm {dist} \left(\varphi (t,\vartheta _{-t}\omega )(B),{\mathcal {A}}(\omega )\right)=0$ almost surely.

There is a slight abuse of notation in the above: the first use of "dist" refers to the Hausdorff semi-distance from a point to a set,

$\mathrm {dist} (x,A):=\inf _{a\in A}d(x,a),$ whereas the second use of "dist" refers to the Hausdorff semi-distance between two sets,

$\mathrm {dist} (B,A):=\sup _{b\in B}\inf _{a\in A}d(b,a).$ As noted in the previous section, in the absence of noise, this definition of attractor coincides with the deterministic definition of the attractor as the minimal compact invariant set that attracts all bounded deterministic sets.

## Theorems relating omega-limit sets to attractors

### The attractor as a union of omega-limit sets

If a random dynamical system has a compact random absorbing set $K$ , then the random global attractor is given by

${\mathcal {A}}(\omega )={\overline {\bigcup _{B}\Omega _{B}(\omega )}},$ ### Bounding the attractor within a deterministic set

$\mathbb {P} \left({\mathcal {A}}(\cdot )\subseteq D\right)>0,$ 