Mixing (mathematics)

The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems.

To provide the mathematical rigor, such descriptions begin with the definition of a measure-preserving dynamical system, written as ⁠

is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, etc.

⁠; this doesn't quite work, as not all subsets of a space have a volume (famously, the Banach–Tarski paradox).

It is always taken to be a Borel set—the collection of subsets that can be constructed by taking intersections, unions and set complements; these can always be taken to be measurable.

A formal difficulty arises when one tries to reconcile the volume of sets with the need to preserve their size under a map.

Mixing asks for this ergodic property to hold between any two sets ⁠

The general situation remains cloudy: for example, given three sets ⁠

The concept of strong mixing is made in reference to the volume of a pair of sets.

of colored dye that is being mixed into a cup of some sort of sticky liquid, say, corn syrup, or shampoo, or the like.

Practical experience shows that mixing sticky fluids can be quite hard: there is usually some corner of the container where it is hard to get the dye mixed into.

One phrases the definition of strong mixing as the requirement that The time parameter

be a measure-preserving dynamical system, with T being the time-evolution or shift operator.

, one has For shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, with

Given a topological space, such as the unit interval (whether it has its end points or not), we can construct a measure on it by taking the open sets, then take their unions, complements, unions, complements, and so on to infinity, to obtain all the Borel sets.

In most applications of ergodic theory, the underlying space is almost-everywhere isomorphic to an open subset of some

Compare this with base in topology, which is less restrictive as it allows non-disjoint unions.

Since the system is assumed to be measure preserving, this last line is equivalent to saying that the covariance ⁠

A system which is strong k-mixing for all k = 2,3,4,... is called mixing of all orders.

Irrational rotations of the circle, and more generally irreducible translations on a torus, are ergodic but neither strongly nor weakly mixing with respect to the Lebesgue measure.

⁠, we can impose a discrete probability distribution on it, then consider the probability distribution on the "coin flip" space, where each "coin flip" can take results from ⁠

In both cases, the shift map (one letter to the left) is strongly mixing, since it is strongly mixing on the covering family of cylinder sets.

A form of mixing may be defined without appeal to a measure, using only the topology of the system.

is said to be topologically transitive if, for every pair of non-empty open sets

⁠, with g being the continuous parameter, with the requirement that a non-empty intersection hold for all ⁠

; more colloquially, the process, in a strong sense, forgets its history.

denote the space of Borel-measurable functions that are square-integrable with respect to the measure

For a stationary Markov process, the coefficients ρt may either decay at an exponential rate, or be always equal to one.

However, when ρt = 1, the process may still be α-mixing, with sub-exponential decay rate.

[3] A strictly stationary Markov process is β-mixing if and only if it is an aperiodic recurrent Harris chain.