Algebraic random walks in the setting of symmetric functions
Section snippets
Introduction and motivation
The subject of algebraic random walks (ARWs) is a development of recent interest in the subject of ‘quantum probability′ [1, 2]. The latter has origins in the foundations of quantum mechanics, but more recently has been influenced by trends towards noncommutative probability. ARWs use a very general algebraic framework [3, 4, 5], in formulating a variety of stochastic processes such as Markov and Lévy processes [6], via a Hopf algebraic formulation [7]. Early studies used commutative function
The ARW formalism
We work in the algebraic framework as developed especially via the study of quantum probability [3], although our current application to standard symmetric functions will of course be in a commutative context. Generically, random variables will be appropriate linear functionals on some operator *-algebra H, and probability measures are provided by states, that is, suitable normalised linear functionals. We are concerned here to develop the consequences of additional co-algebraic structure
Lemma
For a scalar product 〈· | ·〉H: H ⊗ H → ℂ which is compatible with the multiplication in H (see below), then the linear functional ρ: H → ℂ defined by ρ(·) = 〈ρ | ·〉H is positive, 〈ρ | f2〉 ≥ 0 for arbitrary f ∈ H with ρ(f) ∈ ℝ, provided ρ is group-like, Δ(ρ) = ρ ⊗ ρ, or is a positive (convex) combination of group-like elements.
The property is trivially verified using the compatibility property,
Positivity (and normalisation) are clearly maintained by
The Hopf algebra of symmetric functions
We introduce and briefly review the salient features of symmetric functions needed for our purposes.
ARWs and symmetric functions
We wish to apply the ARW construction to the ring of symmetric functions itself, using its Hopf and co-algebraic structures, and to explore the nature of the resulting random walks. While it can be expected that the results will be analogous to those for the standard random walks on the line, the nature of the Hopf- and co-algebras involved is much richer, and provides fertile ground for new types of walks, diffusions and master equations.
In this spirit of ‘experimental’ investigation, we take
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