SHIGEKAWA Ichiro KYOTO UNIVERSITY, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00127234)
HINO Masaomi KYOTO UNIVERSITY, Graduate School of Informatics, Associate Professor, 大学院・情報学研究科, 助教授 (40303888)
KUMAGAI Takashi Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (90234509)
HIGUCHI Yasuko Showa Univ., Coll. of General Education, Associate Professor, 教養部, 講師 (20286842)
SHIRAI Tomoyuki Kanazawa Univ., Fac. of Sciences, Lecturer, 理学部, 助教授 (70302932)
Large deviation principle is one of the most basic laws in probability theory as well as the law of large numbers and the central limit theorem. We have studied the various features of the large deviation principle, starting from the elucidation of the structure of a few new classes of stochastic processes.
We(Shirai and Takahashi) introduced classes of random point fields or point processes, parameterized by a real number α, including fermion point processes(α= -1) and boson point processes(α= 1). They are associated with Fredholm determinants (to the power 1/α)of symmetric integral operators. We established the existence theorem of such random point fields for α= -1/n(n = 1,2,....) and for α = 2/m(M = 1, 2,...) by constructing them in a probabilistic manner. We also proposed a-statistics generalizing. Fermi and Bose statistics and a conjecture that such random fields exist for other values of a. Moreover, w answered to the question raised by Spohn, Johanson and others affirmatively byy showing that such random point fields do exist even for nonsymmetric integral operators provided that they are transition operators of one dimensional diffusions or birth and death processes. Based upon these facts we have established the large deviation principle in addition to other basic limit theorems and ergodic properties such as the estimates of metric entropy and Bernoulli and other properties. These results are published in Ann. Probability, J. Functional Analysis and, ASPM Series vol. 39. Besides them Shirai published an interesting application to Glauber dynamics.
Other investigators have obtained their results related to the large deviation on their own fields: Higuchi (with Shirai) on the random walk and the Schrodinger operator on infinite graphs, Kumagai on the diffusions on fractals, Shigekawa and Hino on the Wiener space and Hara on quadratic Wiener functionals.