Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 046, 21 pages      arXiv:2311.08763      https://doi.org/10.3842/SIGMA.2024.046

Intertwinings for Continuum Particle Systems: an Algebraic Approach

Simone Floreani a, Sabine Jansen b and Stefan Wagner b
a)  Institute for Applied Mathematics, University of Bonn, Bonn, Germany
b)  Mathematisches Institut, Ludwig-Maximilians-Universität, 80333 München, Germany
c)  Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany

Received November 16, 2023, in final form May 22, 2024; Published online June 05, 2024

Abstract
We develop the algebraic approach to duality, more precisely to intertwinings, within the context of particle systems in general spaces, focusing on the $\mathfrak{su}(1,1)$ current algebra. We introduce raising, lowering, and neutral operators indexed by test functions and we use them to construct unitary operators, which act as self-intertwiners for some Markov processes having the Pascal process's law as a reversible measure. We show that such unitaries relate to generalized Meixner polynomials. Our primary results are continuum counterparts of results in the discrete setting obtained by Carinci, Franceschini, Giardinà, Groenevelt, and Redig (2019).

Key words: algebraic approach to stochastic duality; intertwining; inclusion process; Lie algebra $\mathfrak{su}(1,1)$; orthogonal polynomials.

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