Abstract
We obtain stochastic duality functions for specific Markov processes using representation theory of Lie algebras. The duality functions come from the kernel of a unitary intertwiner between *-representations, which provides (generalized) orthogonality relations for the duality functions. In particular, we consider representations of the Heisenberg algebra and mathfrak {su}(1,1). Both cases lead to orthogonal (self-)duality functions in terms of hypergeometric functions for specific interacting particle processes and interacting diffusion processes.
Highlights
A very useful tool in the study of stochastic Markov processes is duality, where information about a specific process can be obtained from another, dual, process
Two processes are in duality if there exists a duality function, i.e. a function of both processes such that the expectations with respect to the original process is related to the expectations with respect to the dual process
In [5,16] orthogonal polynomials of hypergeometric type were obtained as duality functions for several families of stochastic processes, where the orthogonality is with respect to the corresponding stationary measures
Summary
A very useful tool in the study of stochastic Markov processes is duality, where information about a specific process can be obtained from another, dual, process. In [5,16] orthogonal polynomials of hypergeometric type were obtained as duality functions for several families of stochastic processes, where the orthogonality is with respect to the corresponding stationary measures. With a similar method they obtain Bessel functions, which are not polynomials, as self-duality function for a continuous process. In this paper we consider a similar construction with unitary intertwiners between ∗-representations, so that the duality functions will satisfy (generalized) orthogonality relations. 4 we consider discrete series representation of su(1, 1), and obtain Meixner polynomials, Laguerre polynomials and Bessel functions as (self)-duality functions for the symmetric inclusion process and the Brownian energy process. We would like to point out that the self-duality functions are essentially the (generalized) matrix elements for a change of base between bases on which elliptic or parabolic Lie group / algebra elements act diagonally, see e.g. We would like to point out that the self-duality functions are essentially the (generalized) matrix elements for a change of base between bases on which elliptic or parabolic Lie group / algebra elements act diagonally, see e.g. [2,14], so in these cases stochastic self-duality is a consequence of a change of bases in the representation space
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