Major research activity
My research area is the theory of probability and stochastic processes. Probablity theory is theory about random phenomena. The simplest example is coin-tossing. If you toss a coin and the result is the head, take a step to the right, and if it is the tail, then take a step to the left. Repeat this tossing and stepping right on left. Then where are you going to be after tossing the coin many times? How far are you from the starting after you toss the coin ‘n’ times? Are you sure you come back to the starting point again if you continue tossing? How fregments have you spent on the right side? These are the questions about the one dimensional random walk. Although these questions seem quite simple, the methods for answering them are applicable to many situations. For example, the price of a stock company behaves like a random walk, and so the probability theory plays a crucial part when analyzing financial markets. Another example is the study of the materials; because the macroscopic feature of a material is the result of the accumulation of the microscopic behavior, this study can be considered the sum of many coin tossings. I have been interested in the rare events in such random pheonmena. The study of the probability of the rare event is called the theory of large deviations. A major breakthrough was made in the 1920s by Cramer who worked for an insurance company. Another breakthrough came in 1970s with a series of works by Donsker and Varadhan who established the theory of the large deviation principle. I have worked on various models of probability and studied their large deviations.
Major relevant publications
- T. Chiyonobu, “A Limit Formula for a Class of Gibbs Measures with Long Range Pair Interactions,” Journal of Mathematical Sciences The University of Tokyo, 7, 463-486
- T. Chiyonobu, “Large Deviations and Mathematical Physics,” Suuri-Kagaku(Mathematical Science), December 2008, 40-46
- K. Ichihara, T. Chiyonobu, H. Mitsui, “Large Deviation for Periodic Markov Process on Square Lattice,” Journal of Mathematical Sciences The University of Tokyo, 13, 525-544
- K. Ichihara, T. Chiyonobu, “Large Deviation Principle for the Pinned Motion of Random Walks”, Journal of Mathematical Sciences The University of Tokyo, 19, 677-697
- T. Chiyonobu, S. Kusuoka, “The Large Deviation Principle for Hypermixing Processes”, Probability Theory and Related Fields, 78, 627-649, 1988