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Transition Functions of Diffusion Processes on the Thoma Simplex

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Functional Analysis and Its Applications Aims and scope

Abstract

The paper deals with a three-dimensional family of diffusion processes on an infinite-dimensional simplex. These processes were constructed by Borodin and Olshanski in 2009 and 2010, and they include, as limit objects, the infinitely-many-neutral-allels diffusion model constructed by Ethier and Kurtz in 1981 and its extension found by Petrov in 2009.

Each process X in our family possesses a unique symmetrizing measure M, called the z-measure. Our main result is that the transition function of X has a continuous density with respect to M. This is a generalization of earlier results due to Ethier (1992) and to Feng, Sun, Wang, and Xu (2011). Our proof substantially uses a special basis in the algebra of symmetric functions, which is related to the Laguerre polynomials.

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Acknowledgments

I would like to express my gratitude to Grigori Olshanski for suggestions and remarks.

Funding

This work was funded by the Russian Academic Excellence Project’ 5–100.’

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Authors and Affiliations

  1. National Research University Higher School of Economics, Moscow, Russia

    S. Yu. Korotkikh

  2. Massachusetts Institute of Technology, Cambridge, MA, USA

    S. Yu. Korotkikh

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  1. S. Yu. Korotkikh
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Correspondence to S. Yu. Korotkikh.

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Russian Text © The Author(s), 2020, published in Funktsional’nyi Analiz i Ego Prilozheniya, 2020, Vol. 54, No. 2, pp. 58–77.

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Korotkikh, S.Y. Transition Functions of Diffusion Processes on the Thoma Simplex. Funct Anal Its Appl 54, 118–134 (2020). https://doi.org/10.1134/S0016266320020057

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  • DOI: https://doi.org/10.1134/S0016266320020057

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