A Scattering Theory for Markov Chains

In the operator algebraic formulation of probability theory Markov processes typically appear as perturbations of Bernoulli processes. We develop a scattering theory for this situation. This theory applies to the isomorphism problem between Markov processes and Bernoulli shifts as well as to the description of open quantum systems. Introduction. The need to extend classical probability theory in order to include quantum mechanical phenomena has led to an operator algebraic formulation of probability theory ([AFL], [Par], [Mey], [Küm1,3], [KüMa], [Bia]). Here stationary stochastic processes appear as groups of automorphisms of von Neumann algebras with faithful normal states. It has turned out that stationary Markov processes are typically perturbations of Bernoulli processes (Cf. the ‘quantum Feynman-Kac formula’ of [Acc] or the ‘coupling representation’, of [Küm2,3,4], [KüMa]). It is therefore natural to compare the Markov evolution and the Bernoulli shift, i.e., to develop a scattering theory in the spirit of [LPh]. The aim of the present paper is to start such an investigation. We find manageable criteria for scattering operators to exist. They lead to embeddings of Markov processes into Bernoulli shifts, in good cases to conjugacy, as studied in [KeS] and [FrO]. These embeddings can be viewed as algebraic versions of moving average representations. On the physical side this scattering theory generalises the so-called ‘input-output formalism’ of quantum optics ([WaM]), and makes it possible to describe in stochastic terms, for example, the ‘dynamical Stark effect’ in two-level atoms ([RoM]). This paper is organised as follows. Sections 1 and 2 give the necessary background. Sections 3 and 4 develop criteria for the existence of scattering operators. They are applied to various situations in Sections 5, 6 and 7. §1. Probability spaces and stochastic processes. By a non-commutative probability space we shall mean a pair (A, φ) consisting of a von Neumann algebra A equipped with a faithful normal state φ. In the case that A is commutative it can be represented in the form A = L∞(Ω,Σ, μ) for some probability space (Ω,Σ, μ), where φ : A → C sends f ∈ L∞(Ω,Σ, μ) to its expectation ∫ Ω fdμ. The space (A, φ) becomes a pre-Hilbert space when equipped with the inner product 〈x, y〉φ := φ(x∗y); the topology on A induced by the norm ‖x‖φ := φ(x∗x), agrees on bounded sets of A with the strong operator topology. By an operator T : (A, φ) → (B, ψ) we shall always mean a completely positive linear operator T : A → B mapping 1A to 1B and respecting expectations, i.e., ψ◦T = φ. (We note that all operators on (A, φ) are contractions in the norm ‖·‖φ.) In particular, an automorphism S of (A, φ) is a *-automorphism of A leaving the state φ invariant. In the commutative case, an automorphism S is induced by a measure-preserving transformation σ of (Ω,Σ, μ): S(f)(x) = f(σx). An operator P : (A, φ) → (A, φ) satisfying P 2 = P is called a conditional expectation onto its range PA, which is automatically a von Neumann subalgebra of A (cf. [Küm1]). With respect to the pre-Hilbert space structure it is an orthogonal projection. A stochastic process with values in some probability space (A, φ) is a family of *-homomorphisms (‘random variables’) it : (A, φ) → (Â, φ̂), (t ∈ T), of the probability space onto a larger one. Such a process is called stationary if it = T̂t ◦ i0 for some group of automorphisms (T̂t)t∈T of (Â, φ̂). Writing i := i0 we assume in addition that i admits a left inverse P : (Â, φ̂) → (A, φ), i.e., P ◦ i = Id, so