the degree of Doctor of Science

Probability Theory

My main subject is the probability theory. I study in particular that various mathmatical problems arising from statistial physics. I organized a symposium concerning on these and make this research field active. In our department, we have a regular seminar on probability theory. We invite various people to our seminar. This is a good opportunity for our study.

Research Interests

• Probability Theory
  keyword : Probability Theory, Stochastic Analysis, Homogenization of diffusions, diffusions on fractals, infinite particle systems, interacting Brownian motions, Random Matrices
  1980.04I study diffusions on singular spaces. Typical examples of singular spaces are infinitely many particle systems that are appeared in statistical physcis and fractals. These spaces are quite interesting; nevertheless, it is difficult to study diffusions on these spaces because of the usual method cannot be applied to these spaces..

Academic Activities Awards

• Studies on stochastic dynamics of infinite particle systems with long range interaction and its rigidity
• Stochastic Dynamics and Geometry of infinite-particle systems
• Itô Prize 2013 Hirofumi Osada
  "Interacting Brownian motions in infinite dimensions with logarithmic
  interaction potentials II: Airy random point field" SPA 123 (2013), 813-838.
  This is one of a series of papers by Hirofumi Osada on infinite systems of
  interacting particles with logarithmic interaction potentials.
  Osada’s problem concerns the construction of the natural diffusion
  process associated with certain systems of infinite dimensional
  interacting particles by Dirichlet form techniques. For short range
  potentials there are well-known results, but Osada considers a
  logarithmic interaction potential, where none of the previously existing
  theories applies. In particular, the potential is not of Ruelle type and the
  equilibrium state is no longer a Gibbs measure.
  In this series of papers, Osada introduces the notion of quasi-Gibbs
  property, and uses the logarithmic derivative and the scaling property of
  the potential to construct the diffusion process for the particle system. The
  corresponding infinite dimensional SDE is solved as well. Of particular
  importance are the cases of Dyson's Brownian motion and the Ginibre
  point process, both studied by Osada in the previous parts of the series.
  In this paper, Osada shows how his machinery can be applied to the
  Airy random point field, by giving a new sufficient condition for the
  quasi-Gibbs property. Note that the Airy field arises very naturally as the
  edge behavior of Dyson's Brownian motion and it has a close connection
  to random matrix theory. Due to the spatial inhomogeneity, the Airy field
  presents additional difficulties. It is very impressive that the construction
  of the diffusion is still possible for this field.
  This work of Osada makes a bridge between fundamental breakthroughs
  concerning infinite systems of interacting particles and random matrix
  theory. It provides important new classes of stochastic processes with
  values in the configuration space. For these reasons, the board of the
  SPA journal decides to award the Itô Prize 2013 to this paper.
  

Educational Activities