無限粒子系の確率解析学【基盤研究(S)課題番号16H06338】（長田博文／九州大学大学院数理学研究院）

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Outline of the Research

Team Member's Introduction

This project "Stochastic analysis on infinite particle systems" is adopted as the Grant-in-Aid for Scientific Research (S) 16H06338 by Japan Society for the Promotion of Science.

We introduce members of our research team.

Principal Investigator

Co-Investigators

  • Makoto Katori (Faculty of Science and Engineering, Chuo University)
  • Takashi Kumagai (Research Institute for Mathematical Sciences, Kyoto University)
  • Tomohiro Sasamoto (School of Science, Tokyo Institute of Technology)
  • Hideki Tanemura (Department of Mathematics, Keio University)
  • Tadahisa Funaki (School of Fundamental Science and Engineering, Waseda University / Faculty of Mathematical Sciences, The University of Tokyo)
  • Tomoyuki Shirai (Institute of Mathematics for Industry, Kyushu University / Stochastic Analysis Research Center, Kyushu University)

Other Related Investigators

Postdoctoral Fellows

  • Yuki Tokushige (RIMS, Kyoto University)　Apr. 2019--
  • Yosuke Kawamoto　Apr. 2018--Sep. 2018
  • Lu Xu 　Apr. 2017--Mar. 2018
  • Satoshi Yokoyama　Apr. 2017--Mar. 2018
  • Syota Esaki 　Sep. 2016--Mar. 2017
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Purpose of the Research

Infinite particle systems are ensembles of infinite particles with one- or finite-types. They are typical subjects in statistical physics. We consider them as an element of a configration space (a space of a Radon measure consisting of a sum of delta measure), and express its equilibrium state in a point process (a probability measure of the configration space). Moreover, Stochastic dynamics is described as infinite dimensional stochastic differntial equations (ISDEs) with symmetry, in case of random time evolution of continuous motion of infinite many particles labeled in an initial state in the Euclidean space. These are the Stochastic dynamics called intercting Brownian motions (IBMs) in infinite dimensions.
　(For IBMs, please refer to a site of a previous project.)
http://www2.math.kyushu-u.ac.jp/~osada/outline.html (Japanese only)

We have studied ISDEs describing infinite particle systems by developing new methods of analizing scheme and tail events. We will aim at completing it and constructing a new theory to establish an infinite dimensional stochastic analysis.

At the same time, we will study by combining stochastic analysis with the structure of the solvable model known for special subclasses. This analytical method is extremely robust, and it seems that it can be applied not only to SDEs on continuous space but also to discrete space typified by lattice gas, or infinite particle systems with jump-type time evolution. We will pursue expanding a range of its application. We frequently use classical stochastic analyses, such that the Itô analysis and the Dirichlet form theory at the present. We will further connect these with new others such as the Malliavin calculus, the Rough path theory, Stochastic partial differential equations and so on.

Focusing on one particle in infinite particle systems, it becomes a problem of random medium. We would like to tackle interesting problems motivated by various statistical physics such as KPZ equation, Homogenization problem and Phase transition phenomenon.

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Main Results

The starting point of this research is the following four-part papers 1.--4., published by the Principal investigator. We have constructed a general theory to solve ISDEs describing IBMs there.

  1. H. Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials. Ann. Probab. 41 (2013), no. 1, 1–49.
  2. H. Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II: Airy random point field. Stochastic Process. Appl. 123 (2013), no. 3, 813–838.
  3. H. Osada, Infinite-dimensional stochastic differential equations related to random matrices. Probab. Theory Related Fields 153 (2012), no. 3-4, 471–509.
  4. H. Osada, Tagged particle processes and their non-explosion criteria. J. Math. Soc. Japan 62 (2010), no. 3, 867–894.

We call it a first theory for convenience. This can not only be obviously applied to all Gibbs measures (essentially), but it can also be applied to RPFs with logarithmic potentials and other long-distance strong interactions that were previously impossible. One of main purposes of this research project was to adapt it to the Airy RPFs. In doing so, we built another general theory to prove the existence and pathwise uniqueness of strong solutions of ISDEs with high symmetry, represented by IBMs, with Co-Investigator Hideki Tanemura.
The basic idea of the proof is to analyze tail events and to reconsider ISDEs as infinite number of finite SDEs schemes. We pridefully believe that it is a novel method. In particular, we obtain a wide variety of very strong results that were not considered before, by showing the uniqueness of the solution of ISDEs. It seems that it is far beyond the expectatioon at first. Regarding the two main results, the second general theory is published in the following paper No.5.

  1. H. Osada, H. Tanemura, Infinite-dimensional stochastic differential equations and tail σ-fields, Probability Theory and Related Fields (2020). https://doi.org/10.1007/s00440-020-00981-y http://arxiv.org/abs/1412.8674v12
  2. Y. Kawamoto, H. Osada, H. Tanemura, Infinite-dimensional stochastic differential equations and tail σ-fields II: the IFC condition, https://arxiv.org/abs/2007.03214

An application to the Airy RPFs (Airy IBMs) is published in the following paper No.6.

  1. H. Osada, H. Tanemura, Infinite-dimensional stochastic differential equations arising from Airy random point field http://arxiv.org/abs/1408.0632v5  (preprint)

We refer to the general theory that started with paper No.5 as the second theory. It seems to produce various spin-offs. Indeed, seven papers are recently published and paper No.7 will be scheduled for publication.

  1. Alexander I Bufetov, Andrey V Dymov, Hirofumi Osada
    The logarithmic derivative for point processes with equivalent Palm measures
    to appear in JMSJ. http://arxiv.org/abs/1707.01773
  2. Y. Kawamoto, H. Osada
    Finite particle approximations of interacting Brownian particles with logarithmic potentials
    J. Math. Soc. Japan, Volume 70, Number 3 (2018), 921-952. doi:10.2969/jmsj/75717571
    https://projecteuclid.org/euclid.jmsj/1529309020 PDF File
  3. Yosuke Kawamoto, Hirofumi Osada
    Dynamical Bulk Scaling limit of Gaussian Unitary Ensembles and Stochastic-Differential-Equation gaps, Journal of Theoretical Probability (2018).
    https://doi.org/10.1007/s10959-018-0816-2 http://arxiv.org/abs/1610.05969v2
  4. Hirofumi Osada, Shota Osada
    Discrete approximations of determinantal point processes on continuous spaces: tree representations and tail triviality, Journal of Statistical Physics, 170(2), 421--435.
    Jan. 2018, https://doi.org/10.1007/s10955-017-1928-2
  5. H. Osada, H. Tanemura, Strong Markov property of determinantal processes with extended kernels. Stochastic Processes and their Applications 126 (1), 186-208 (2016), DOI 10.1016/j.spa.2015.08.003
  6. H. Osada, T. Shirai, Absolute continuity and singularity of Palm measures of the Ginibre point process. Probability Theory and Related Fields 165 (3-4), 725-770 (2016), DOI 10.1007/s00440-15-0644-6
  7. R. Honda, H. Osada, Infinite-dimensional stochastic differential equations related to Bessel random point fields. Stochastic Processes and their Applications 125 (2015), no. 10, 3801–3822
  8. H. Osada, H. Tanemura, Cores of Dirichlet forms related to random matrix theory. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 10, 145–150.

Besides these, we are preparing many papers on this research, and that shows a big progression.
The explanations of these general theories, "Stochastic analysis for infinite many particle systems: --Random matrices and intercting Brownian motions in infinite dimensions", is posted in "Sūgaku" and will be "Sugaku Expositions".

PDF file

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