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The Workshop is finantialy supported by
• Faculty of Pure and Applied Mathematics, Wrocław University of Scince and Technology
• Grant-in-Aid for Scientific Research (C) (No.26400405) of Japan Society for the Promotion of Science
• The European Union Programme Erasmus+
Workshop will be held at Faculty of Pure and Applied Mathematics - room P.01 bulding C-11 (14a. Janiszewskiego St.)
PROGRAMME:
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27 June 2018 (Wednesday)
• 9:15 - 10:15 - MAKOTO KATORI (Chuo University, Faculty of Science and Engineering, JAPAN)
"Limit theorems for interacting Brownian motions I"
ABSTRACT: Dyson model is a stochastic particle system in one dimension R, in which repulsive force acts between any pair of particles with strength proportional to the inverse of distance. This multivariate stochastic process is realized as the system of one-dimensional Brownian motions conditioned never to collide with each other. We can show that this many-body system is exactly solvable and of determinantal in the sense that any spatio-temporal correlation function is expressed by determinant and is controlled by a single continuous function called the correlation kernel. In this lecture, we assume the special initial configuration such that all N particles are concentrated on the origin and we discuss the limit theorems in N ⟶ ∞.
Wigner's semicircle law, which is extensively studied in random matrix theory and free probability, is demonstrated
as the law of large numbers (LLN), which describes the density profile of particles in R at each time. Two kinds of limits called the bulk scaling limit and the soft-edge scaling limit are introduced in order to obtain determinantal processes with an infinite number of particles. As the central limit theorem (CLT) associated with the latter scaling limit, the Tracy--Widom distribution is discussed.
Lecture Notes [PDF]
• 10:30 - 11:45 - PIOTR GRACZYK (University of Angers, LAREMA, FRANCE)
"Solving SDEs for Squared Bessel particle systems"
ABSTRACT:
We study the existence, uniqueness, collisions and other properties of
Squared Bessel particle systems in full generality, admitting
any drift parameter α∈R and allowing that
the particle positions take also negative values. We extend the results obtained by Going-Jaeschke and Yor in 1-dimensional case.
There are two difficulties to overcome in solving SDEs for Squared Bessel particle systems:
1. the martingale parts √X_idW_i are non-Lipschitz.
However our multidimensional Yamada-Watanabe theorem proven
in 2013 does not cover colliding starting points
2. the drift parts contain singular denominators X_i-X_j,
like Dyson BM.
Squared Bessel particle systems may be interpreted as eigenvalues of generalized Wishart processes. We determine the parameter set of Wishart processes, i.e. the stochastic Gindikin set.
Our techniques use elementary symmetric polynomials and their SDEs systems. These are joint works with J. Malecki, E. Mayerhofer and K. Bogus.
28 June 2018 (Thursday)
• 9:15 - 10:15 - MAKOTO KATORI (Chuo University, Faculty of Science and Engineering, JAPAN)
"Limit theorems for interacting Brownian motions II"
ABSTRACT: Second part.
• 10:30 - 11:45 - MAKOTO KATORI (Chuo University, Faculty of Science and Engineering, JAPAN)
"Limit theorems for interacting Brownian motions III"
ABSTRACT: Third part.
• 12:00 - 12:30 - KAMIL BOGUS (University of Angers, LAREMA, FRANCE)
"On beta-versions of the squared Bessel particle systems."
• 12:30 - lunch break
• 15:15 - Open problem session and discussions
29 June 2018 (Friday) room 2.11 bulding C-11
• 9:15 - 10:30 - JACEK MAŁECKI (WUST, Faculty of Pure and Applied Mathematics, POLAND)
"Convergence of empirical measures for eigenvalues of general matrix-valued processes"
ABSTRACT: We will discuss solutions to general matrix-valued SDEs on the space of symmetric and Hermitian matrices. We will find the stochastic description of the related empirical measure-valued processes and show their weak convergence to certian families of measures. The limiting families are described by integral equations related to coefficients of the initial matrix SDE. Our approach is based on the basic symmetric polynomials and power sums of the eigenvalues processes.
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