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Stochastic Duality and Orthogonal Polynomials

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Sojourns in Probability Theory and Statistical Physics - III

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 300))

Abstract

For a series of Markov processes we prove stochastic duality relations with duality functions given by orthogonal polynomials. This means that expectations with respect to the original process (which evolves the variable of the orthogonal polynomial) can be studied via expectations with respect to the dual process (which evolves the index of the polynomial). The set of processes include interacting particle systems, such as the exclusion process, the inclusion process and independent random walkers, as well as interacting diffusions and redistribution models of Kipnis–Marchioro–Presutti type. Duality functions are given in terms of classical orthogonal polynomials, both of discrete and continuous variable, and the measure in the orthogonality relation coincides with the process stationary measure.

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Acknowledgments

This research was supported by the Italian Research Funding Agency (MIUR) through FIRB project grant no. RBFR10N90W and in part by the National Science Foundation under Grant No. NSF PHY11-25915. We acknowledge a useful discussion on the topic of this paper with Cédric Bernardin during the trimester “Disordered systems, random spatial processes and their applications” that was held at the Institute Henri Poincaré.

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Authors and Affiliations

  1. Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, 1049-001, Lisbon, Portugal

    Chiara Franceschini

  2. University of Modena and Reggio Emilia, via G. Campi 213/b, 41125, Modena, Italy

    Cristian Giardinà

Authors
  1. Chiara Franceschini
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  2. Cristian Giardinà
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Correspondence to Cristian Giardinà .

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Editors and Affiliations

  1. NYU Shanghai, Shanghai, China

    Vladas Sidoravicius

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To Charles M. Newman on his 70 $$^\mathrm{th}$$ birthday

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Franceschini, C., Giardinà, C. (2019). Stochastic Duality and Orthogonal Polynomials. In: Sidoravicius, V. (eds) Sojourns in Probability Theory and Statistical Physics - III. Springer Proceedings in Mathematics & Statistics, vol 300. Springer, Singapore. https://doi.org/10.1007/978-981-15-0302-3_7

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