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Mixing (mathematics)

Mixing (mathematics)

In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, etc.

The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including strong mixing, weak mixing and topological mixing, with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" notion than ergodicity).

Mixing in stochastic processes

Letbe astochastic processon a probability space. The sequence space into which the process maps can be endowed with a topology, theproduct topology. Theopen setsof this topology are calledcylinder sets. These cylinder sets generate aσ-algebra, theBorel σ-algebra; this is the smallest σ-algebra that contains the topology.
Define a function, called the strong mixing coefficient, as
for all. The symbol, withdenotes a sub-σ-algebra of the σ-algebra; it is the set of cylinder sets that are specified between times a and b, i.e. the σ-algebra generated by.
The processis said to be strongly mixing ifas. That is to say, a strongly mixing process is such that, in a way that is uniform over all timesand all events, the events before timeand the events after timetend towards beingindependentas; more colloquially, the process, in a strong sense, forgets its history.

Types of mixing

Supposewere a stationaryMarkov processwith stationary distributionand letdenote the space of Borel-measurable functions that are square-integrable with respect to the measure. Also let
denote the conditional expectation operator onFinally, let

denote the space of square-integrable functions with mean zero.

The ρ of the process {xt} are

The process is called ρ if these coefficients converge to zero as t → ∞, and “ρ-mixing with exponential decay rate” if ρt < e−δt for some δ > 0. For a stationary Markov process, the coefficients ρt may either decay at an exponential rate, or be always equal to one.[1]

The α of the process {xt} are

The process is called α if these coefficients converge to zero ast → ∞, it is “α-mixing with exponential decay rate” if*αt
  • < γe
−δtfor someδ > 0, and it is α-mixing with a sub-exponential decay rate if*αt
  • < ξ(t)
for some non-increasing functionsatisfying
as.[1]

The α-mixing coefficients are always smaller than the ρ-mixing ones: αt ≤ ρt, therefore if the process is ρ-mixing, it will necessarily be α-mixing too. However, when ρt = 1, the process may still be α-mixing, with sub-exponential decay rate.

The β are given by

The process is called β if these coefficients converge to zero ast → ∞, it is β-mixing with an exponential decay rate if*βt
  • < γe
−δtfor someδ > 0, and it is β-mixing with a sub-exponential decay rate if*βtξ*(t) → 0ast → ∞for some non-increasing functionsatisfying
as.[1]

A strictly stationary Markov process is β-mixing if and only if it is an aperiodic recurrent Harris chain. The β-mixing coefficients are always bigger than the α-mixing ones, so if a process is β-mixing it will also be α-mixing. There is no direct relationship between β-mixing and ρ-mixing: neither of them implies the other.

Mixing in dynamical systems

A similar definition can be given using the vocabulary ofmeasure-preserving dynamical systems. Letbe a dynamical system, with T being the time-evolution orshift operator. The system is said to be strong mixing if, for any, one has
For shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, withreplaced bywith g being the continuous-time parameter.
To understand the above definition physically, consider a shakerfull of an incompressible liquid, which consists of 20% wine and 80% water. Ifis the region originally occupied by the wine, then, for any regionwithin the shaker, the percentage of wine inafterrepetitions of the act of stirring is
In such a situation, one would expect that after the liquid is sufficiently stirred (), every regionof the shaker will contain approximately 20% wine. This leads to
where, because measure-preserving dynamical systems are defined on probability spaces, and hence the final expression implies the above definition of strong mixing.

A dynamical system is said to be weak mixing if one has

In other words,is strong mixing ifin the usual sense, weak mixing if
in theCesàrosense, and ergodic ifin the Cesàro sense. Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converse is not true: there exist ergodic dynamical systems which are not weakly mixing, and weakly mixing dynamical systems which are not strongly mixing.

For a system that is weak mixing, the shift operator T will have no (non-constant) square-integrable eigenfunctions with associated eigenvalue of one. In general, a shift operator will have a continuous spectrum, and thus will always have eigenfunctions that are generalized functions. However, for the system to be (at least) weak mixing, none of the eigenfunctions with associated eigenvalue of one can be square integrable.

formulation

The properties of ergodicity, weak mixing and strong mixing of a measure-preserving dynamical system can also be characterized by the average of observables. By von Neumann's ergodic theorem, ergodicity of a dynamical systemis equivalent to the property that, for any function, the sequenceconverges strongly and in the sense of Cesàro to, i.e.,
A dynamical systemis weakly mixing if, for any functionsand
A dynamical systemis strongly mixing if, for any functionthe sequenceconverges weakly toi.e., for any function
Since the system is assumed to be measure preserving, this last line is equivalent to saying thatso that the random variablesandbecome orthogonal asgrows. Actually, since this works for any functionone can informally see mixing as the property that the random variablesandbecome independent asgrows.

Products of dynamical systems

Given two measured dynamical systemsandone can construct a dynamical systemon the Cartesian product by definingWe then have the following characterizations of weak mixing:
Proposition. A dynamical systemis weakly mixing if and only if, for any ergodic dynamical system, the systemis also ergodic.
Proposition. A dynamical systemis weakly mixing if and only ifis also ergodic. If this is the case, thenis also weakly mixing.

Generalizations

The definition given above is sometimes called strong 2-mixing, to distinguish it from higher orders of mixing. A strong 3-mixing system may be defined as a system for which

holds for all measurable sets A, B, C. We can define strong k-mixing similarly. A system which is strong k-mixing for all k = 2,3,4,... is called mixing of all orders.

It is unknown whether strong 2-mixing implies strong 3-mixing. It is known that strong m-mixing implies ergodicity.

Examples

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.

Many maps considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the dyadic map, Arnold's cat map, horseshoe maps, Kolmogorov automorphisms, and the geodesic flow on the unit tangent bundle of compact surfaces of negative curvature.

Topological mixing

A form of mixing may be defined without appeal to ameasure, using only thetopologyof the system. Acontinuous mapis said to be topologically transitive if, for every pair of non-emptyopen sets, there exists an integer n such that
whereis thenth iterateof f. In theoperator theory, a topologically transitivebounded linear operator(a continuous linear map on atopological vector space) is usually calledhypercyclic operator. A related idea is expressed by thewandering set.
Lemma: If X is acompletemetric spacewith noisolated point, then f is topologically transitive if and only if there exists ahypercyclic point, that is, a point x such that its orbitisdensein X.
A system is said to be topologically mixing if, given open setsand, there exists an integer N, such that, for all, one has
For a continuous-time system,is replaced by theflow, with g being the continuous parameter, with the requirement that a non-empty intersection hold for all.

A weak topological mixing is one that has no non-constant continuous (with respect to the topology) eigenfunctions of the shift operator.

Topological mixing neither implies, nor is implied by either weak or strong mixing: there are examples of systems that are weak mixing but not topologically mixing, and examples that are topologically mixing but not strong mixing.

References

[1]
Citation Linkopenlibrary.orgChen, Xiaohong; Hansen, Lars Peter; Carrasco, Marine (2010). "Nonlinearity and temporal dependence". Journal of Econometrics. 155 (2): 155–169. CiteSeerX 10.1.1.597.8777. doi:10.1016/j.jeconom.2009.10.001.
Sep 29, 2019, 9:56 PM
[2]
Citation Linkciteseerx.ist.psu.edu10.1.1.597.8777
Sep 29, 2019, 9:56 PM
[3]
Citation Linkdoi.org10.1016/j.jeconom.2009.10.001
Sep 29, 2019, 9:56 PM
[4]
Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
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