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.
Consider a random dynamical system formula_1 on a complete separable metric space formula_2, where the noise is chosen from a probability space formula_3 with base flow formula_4.
A naïve definition of an attractor formula_5 for this random dynamical system would be to require that for any initial condition formula_6, formula_7 as formula_8. This definition is far too limited, especially in dimensions higher than one. A more plausible definition, modelled on the idea of an omega-limit set, would be to say that a point formula_9 lies in the attractor formula_5 if and only if there exists an initial condition, formula_6, and there is a sequence of times formula_12 such that
This is not too far from a working definition. However, we have not yet considered the effect of the noise formula_15, 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 formula_16 seconds into the "future", and considering the limit as formula_8, one "rewinds" the noise formula_16 seconds into the "past", and evolves the system through formula_16 seconds using the same initial condition. That is, one is interested in the pullback limit
So, for example, in the pullback sense, the omega-limit set for a (possibly random) set formula_21 is the random set
Equivalently, this may be written as
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.
Several examples of pullback attractors of non-autonomous dynamical systems are presented analytically and numerically.
Definition.
The pullback attractor (or random global attractor) formula_24 for a random dynamical system is a formula_25-almost surely unique random set such that
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,
whereas the second use of "dist" refers to the Hausdorff semi-distance between two sets,
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 formula_38, then the random global attractor is given by
where the union is taken over all bounded sets formula_34.
Bounding the attractor within a deterministic set.
Crauel (1999) proved that if the base flow formula_41 is ergodic and formula_42 is a deterministic compact set with
then formula_44 formula_25-almost surely.