joinings

Joinings are a concept in probability theory and ergodic theory that describe how two stochastic processes or dynamical systems can be coupled together while preserving their individual distributions. They formalize the idea of building a joint system that exhibits a prescribed dependence structure between its components.

Formally, take two measure-preserving dynamical systems (X, B, μ, T) and (Y, C, ν, S) on probability spaces. A joining λ of these systems is a probability measure on the product space X×Y such that its marginals are μ and ν (i.e., the projections recover μ and ν) and it is invariant under the product transformation T×S: (T×S)_* λ = λ. When one ignores dynamics and only asks for a static coupling of the stationary distributions, a joining is simply a coupling of μ and ν.

Examples: The product measure μ×ν on X×Y is a joining and represents independence between the two systems. If the two systems are the same (X, B, μ, T) = (Y, C, ν, S), the diagonal measure Δ defined by Δ(A×B) = μ(A ∩ B) is a joining, corresponding to perfect agreement (x is paired with itself) under T×T.

Typical topics: The set of joinings of μ and ν is convex and compact in the weak topology. Ergodic joinings are the extreme points. Self-joinings (joining a system with itself) are studied to understand internal structure. Disjointness means that the only joining is the product μ×μ, indicating maximal independence.

Applications: Joinings are used to compare dynamical systems, study mixing and ergodic properties, and construct couplings in probability theory.