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. 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). Let ( X t ) − ∞ < t < ∞ {displaystyle (X_{t})_{-infty <t<infty }} be a stochastic process on a probability space ( Ω , F , P ) {displaystyle (Omega ,{mathcal {F}},mathbb {P} )} . The sequence space into which the process maps can be endowed with a topology, the product topology. The open sets of this topology are called cylinder sets. These cylinder sets generate a σ-algebra, the Borel σ-algebra; this is the smallest σ-algebra that contains the topology. Define a function α {displaystyle alpha } , called the strong mixing coefficient, as for all − ∞ < s < ∞ {displaystyle -infty <s<infty } . The symbol X a b {displaystyle X_{a}^{b}} , with − ∞ ≤ a ≤ b ≤ ∞ {displaystyle -infty leq aleq bleq infty } denotes 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 { X a , X a + 1 , … , X b } {displaystyle {X_{a},X_{a+1},ldots ,X_{b}}} . The process ( X t ) − ∞ < t < ∞ {displaystyle (X_{t})_{-infty <t<infty }} is said to be strongly mixing if α ( s ) → 0 {displaystyle alpha (s) o 0} as s → ∞ {displaystyle s o infty } . That is to say, a strongly mixing process is such that, in a way that is uniform over all times t {displaystyle t} and all events, the events before time t {displaystyle t} and the events after time t + s {displaystyle t+s} tend towards being independent as s → ∞ {displaystyle s o infty } ; more colloquially, the process, in a strong sense, forgets its history. Suppose ( X t ) {displaystyle (X_{t})} were a stationary Markov process with stationary distribution Q {displaystyle mathbb {Q} } and let L 2 ( Q ) {displaystyle L^{2}(mathbb {Q} )} denote the space of Borel-measurable functions that are square-integrable with respect to the measure Q {displaystyle mathbb {Q} } . Also let denote the conditional expectation operator on L 2 ( Q ) . {displaystyle L^{2}(mathbb {Q} ).} Finally, let