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Transition density of an infinite-dimensional diffusion with the jack parameter

Part of: Markov processes Combinatorial probability

Published online by Cambridge University Press:  16 January 2023

Youzhou Zhou
Youzhou Zhou*
Affiliation:
Xi’an Jiaotong-Liverpool University
*
*Postal address: 111 Renai Road, Suzhou, Jiangsu, China 215 123. Email address: Youzhou.zhou@xjtlu.edu.cn
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Abstract

From the Poisson–Dirichlet diffusions to the Z-measure diffusions, they all have explicit transition densities. We show that the transition densities of the Z-measure diffusions can also be expressed as a mixture of a sequence of probability measures on the Thoma simplex. The coefficients are the same as the coefficients in the Poisson–Dirichlet diffusions. This fact will be uncovered by a dual process method in a special case where the Z-measure diffusions are established through an up–down chain in the Young graph.

Keywords

Jack graphtransition densitydual processup–down Markov chainKingman coalescent

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60C05: Combinatorial probability
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 797 - 811
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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