Noncolliding Brownian bridges associated with elliptic functions

Date
2015/03/25 Wed 15:30 - 17:00
Room
3号館552号室
Speaker
Makoto Katori
Affiliation
Chuo University
Abstract

Dyson's Brownian motion model with β=2 was originally introduced as an eigenvalue process of Hermitian-matrix-valued Brownian motion in random matrix theory. Grabiner (1999) constructed it as the Doob h-transform of absorbing Brownian motion in a Weyl chamber and realized it as a system of one-dimensional Brownian motions conditioned never to collide with each other (the noncolliding Brownian motion). By replacing the rational functions appearing in the h-transform with the elliptic functions having time-dependent nomes q=q(t), we introduced an elliptic extension of the Dyson model, which is an inhomogeneous diffusion process of noncolliding particles defined on a circle in a finite time period [0,t), 0<t<. When the process starts from the configuration with equidistant spacing on the circle (denoted by η), we gave a determinantal martingale representation and proved that the process is determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single continuous function called the correlation kernel (to appear in PTRF, DOI 10.1007/s00440-014-0581-9). In the present talk, we put emphasis on the important role of Jacobi's imaginary transformations (the modular transformations) in such `elliptic-functional diffusion processes' and show that our process can be regarded as a system of noncolliding Brownian bridges of length t from η to η. An infinite particle limit will be also discussed.

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