Department Seminars & Colloquia
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Based on the quantum white noise theory, we introduce the new concept of quantum white noise derivatives of white noise operators. As applications we solve implementation problems for the canonical commutation relation and for a quantum extension of Girsanov transformation.
‣ Date & Time : 7/21, 10:00~11:00, 11:10~12:10
In the second lecture we continue the discussion of orthogonal polynomials, now dealing with multi-variable functions. By introducing creation, annihilation, and preservation operators for the multi-variables, we construct again an interacting Fock space (IFS). Thereby we extend the theory of orthogonal polynomials in the 1-dimensional space to that in the multi-dimensional space. As a byproduct we show the relationship between the support of the measure and the deficiency rank of the form generator, which appears in the construction of the IFS. We finish with some open problems. This lecture is based on the joint work with A. Dhahri (Chungbuk) and N. Obata (Tohoku).
- Date & Time : 7/21, 13:30~14:30, 15:00~16:00
We start with the standard construction of generalized white noise functionals as infinite dimensional distributions and we study the analytic characterization theorem for S-transform of generalized white noise functionals. Then we study basic concepts and results on white noise operators which is necessary for the study of quantum white noise calculus. The analytic characterization of operator symbols and the Fock expansion theorem are of particular importance.
In the first part of the lectures we will discuss 1-dimensional orthogonal polynomials. Main topics that will be discussed are the followings.
- Three-term recurrence relation and the Jacobi coefficients
- Examples
- Graph spectrum
- Interacting Fock spaces
- Accardi-Bozejko formula
The main reference for this lecture is <Quantum probability and spectral analysis of graphs>, Springer, 2007, by A. Hora and N. Obata.
‣ Date & Time : 7/20, 13:30~14:30, 15:00~16:00
We proposes a computational framework for continuous time opinion dynamics with additive noise. We derive a non-local partial differential equation for the distribution of opinions differences. We use Mellin transforms to solve the stationary solution of this equation in closed form. This approach can be applied both to linear dynamics on an interaction graph and to bounded confidence dynamics in the Euclidean space. To the best of our knowledge, the closed form expression on the stationary distribution of the bounded confidence model is the first quantitative result on the equilibria of this class of models. The solutions are presented here in the simplest possible cases (small number of agents, present of stubborn agents, one dimensional opinions).
In this talk, we consider stochastic partial differential equations, especially, a parabolic Anderson model. This model shows intermittent phenomena, i.e., the solution becomes very big on small regions of different scales (we say tall peaks occur on small islands). We provide a way to quantify tall peaks and small islands by using the macroscopic fractal dimension theory by Barlow and Taylor. This is based on joint work with Davar Khoshnevisan and Yimin Xiao.