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 unitary 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.
Highlights
In the theory of interacting particle systems [28, 13], and more generally in the theory of Markov processes, stochastic duality plays a key role
A list of examples of systems that have been analyzed using duality includes: boundary driven models of transport and derivation of Fourier’s law [21, 33, 15, 7], diffusive particle systems and their hydrodynamic limit [13], asymmetric interacting particle systems scaling to KPZ equation [31, 30, 6, 12], six vertex models [4, 11], multispecies particle models [24, 25, 26, 2], correlation inequalities [18] and mathematical population genetics [29, 10]
The Symmetric Inclusion Processes (SIP) is a family of Markov jump processes labeled by parameter k > 0, which can be defined in the same setting of before
Summary
In the theory of interacting particle systems [28, 13], and more generally in the theory of Markov processes, stochastic duality plays a key role. In particular the works [3, 16, 27, 5] prove that for a large class of processes duality functions are provided by orthogonal polynomials This result, which has been proved following an analytic approach – either using structural properties of hypergeometric polynomials [16] or generating function methods [27] – has been put in an algebraic perspective in [19], where it is shown that orthogonal duality relations correspond to unitarily equivalent representations. One would like to see some symmetries in the generator of the process: these two facts guarantees that a self-duality relation of the process via the symmetric function can be found This is the main message of this paper that will be formulated in Theorem 2.7. As an application of the theorem we will show that several known self-duality functions can be derived in this way and we will derive a new self-duality relation for the so-called Brownian momentum process [17]
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