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Self-Duality of Markov Processes and Intertwining Functions

  • Chiara FranceschiniEmail author
  • Cristian Giardinà
  • Wolter Groenevelt
Article

Abstract

We present a theorem which elucidates the connection between self-duality of Markov processes and representation theory of Lie algebras. In particular, we identify sufficient conditions such that the intertwining function between two representations of a certain Lie algebra is the self-duality function of a (Markov) operator. In concrete terms, the two representations are associated to two operators in interwining relation. The self-dual operator, which arise from an appropriate symmetric linear combination of them, is the generator of a Markov process. The theorem is applied to a series of examples, including Markov processes with a discrete state space (e.g. interacting particle systems) and Markov processes with continuous state space (e.g. diffusion processes). In the examples we use explicit representations of Lie algebras that are unitarily equivalent. As a consequence, in the discrete setting self-duality functions are given by orthogonal polynomials whereas in the continuous context they are Bessel functions.

Keywords

Stochastic duality Representation theory Lie algebras Orthogonal polynomials 

Mathematics Subject Classification (2010)

60J25 60J27 60J60 82C05 

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.University of FerraraFerraraItaly
  2. 2.University of Modena and Reggio EmiliaModenaItaly
  3. 3.Technische Universiteit Delft, DIAMDelftThe Netherlands

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