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Hirofumi Osada Last modified date：2020.12.02

Professor / Department of Mathematical Sciences
Faculty of Mathematics


Graduate School
Department of Mathematics
Graduate School of Mathematics
Undergraduate School
Department of Mathematics
School of Sciences
Other Organization
Stochastic Analysis Research Center


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Homepage
https://kyushu-u.pure.elsevier.com/en/persons/hirofumi-osada
　Reseacher Profiling Tool　Kyushu University Pure
http://www2.math.kyushu-u.ac.jp/~osada-labo/osadapersonal_hp/index.html
Research, List of papers, preprints and lectures, CV .
http://www2.math.kyushu-u.ac.jp/~osada-labo/center%20HP/index.html
Stochastic Analysis Research Center .
http://www2.math.kyushu-u.ac.jp/~osada-labo/kiban_s_HP/en/index.html
"Stochastic analysis on infinite particle systems", JSPS Kakenhi, KIBAN-S(16H06338) .
Academic Degree
the degree of Doctor of Science
Country of degree conferring institution (Overseas)
No
Field of Specialization
Probability Theory
Total Priod of education and research career in the foreign country
00years00months
Outline Activities
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
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
Papers
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1. Hirofumi Osada, Hideki Tanemura, Infinite-dimensional stochastic differential equations and tail σ-fields, Probability Theory and Related Fields , 10.1007/s00440-020-00981-y, 177, 1137-1242, 2020.07, We present general theorems solving the long-standing problem of the existence and pathwise uniqueness of strong solutions of infinite-dimensional stochastic differential equations (ISDEs) called interacting Brownian motions. These ISDEs describe the dynamics of infinitely-many Brownian particles moving in Rd with free potential Φ and mutual interaction potential Ψ. We apply the theorems to essentially all interaction potentials of Ruelle’s class such as the Lennard-Jones 6-12 potential and Riesz potentials, and to logarithmic potentials appearing in random matrix theory. We solve ISDEs of the Ginibre interacting Brownian motion and the sineβ interacting Brownian motion with β=1,2,4. We also use the theorems in separate papers for the Airy and Bessel interacting Brownian motions. One of the critical points for proving the general theorems is to establish a new formulation of solutions of ISDEs in terms of tail σ-fields of labeled path spaces consisting of trajectories of infinitely-many particles. These formulations are equivalent to the original notions of solutions of ISDEs, and more feasible to treat in infinite dimensions..
2. Alexander I. Bufetov, Andrey V. Dymov, Hirofumi Osada, The logarithmic derivative for point processes with equivalent Palm measures, Journal of the Mathematical Society of Japan, 10.2969/jmsj/78397839, 71, 2, 451-469, 2019.01, The logarithmic derivative of a point process plays a key role in the general approach, due to the third author, to constructing diffusions preserving a given point process. In this paper we explicitly compute the logarithmic derivative for determinantal processes on R with integrable kernels, a large class that includes all the classical processes of random matrix theory as well as processes associated with de Branges spaces. The argument uses the quasi-invariance of our processes established by the first author..
3. Hirofumi Osada, Tomoyuki Shirai, Absolute continuity and singularity of Palm measures of the Ginibre point process, Probability Theory and Related Fields, 10.1007/s00440-015-0644-6, 165, 3-4, 725-770, 2016.08, We prove a dichotomy between absolute continuity and singularity of the Ginibre point process G and its reduced Palm measures { G<sub>x</sub>, x∈ C<sup>ℓ</sup>, ℓ= 0 , 1 , 2 … } , namely, reduced Palm measures G<sub>x</sub> and G<sub>y</sub> for x∈ C<sup>ℓ</sup> and y∈ C<sup>n</sup> are mutually absolutely continuous if and only if ℓ= n; they are singular each other if and only if ℓ≠ n. Furthermore, we give an explicit expression of the Radon–Nikodym density dG<sub>x</sub>/ dG<sub>y</sub> for x, y∈ C<sup>ℓ</sup>..
4. Hirofumi Osada, Hideki Tanemura, Strong Markov property of determinantal processes with extended kernels, Stochastic Processes and their Applications, 10.1016/j.spa.2015.08.003, 126, 1, 186-208, 2016.01, Noncolliding Brownian motion (Dyson's Brownian motion model with parameter β=2) and noncolliding Bessel processes are determinantal processes; that is, their space-time correlation functions are represented by determinants. Under a proper scaling limit, such as the bulk, soft-edge and hard-edge scaling limits, these processes converge to determinantal processes describing systems with an infinite number of particles. The main purpose of this paper is to show the strong Markov property of these limit processes, which are determinantal processes with the extended sine kernel, extended Airy kernel and extended Bessel kernel, respectively. We also determine the quasi-regular Dirichlet forms and infinite-dimensional stochastic differential equations associated with the determinantal processes.
ランダム行列に関係する1次元空間の無限粒子系の運動を記述する確率力学の構成には、確率解析的な手法と、時空間相関関数の計算に基づく代数的な方法がある。この論文は、代数的な手法によって構成された確率力学の強マルコフ性を証明し、筆者達の他の結果と合わせて、これら二つの手法によって構成された確率力学が同一であることを証明した。その結果、解析的手法によって証明されている粒子の運動のパスとしての性質、また、代数的に手法によって証明されている確率力学の定量的な性質が、これらは、それぞれ別の手法ではとr手も証明されないものだが、これら二つの確率力学の同一性が示されたために、両方に対して成立することが判明した。.
5. Hirofumi Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II
Airy random point field, Stochastic Processes and their Applications, 10.1016/j.spa.2012.11.002, 123, 3, 813-838, 2013.01, We give a new sufficient condition of the quasi-Gibbs property. This result is a refinement of one given in a previous paper (Osada (in press) [18]), and will be used in a forthcoming paper to prove the quasi-Gibbs property of Airy random point fields (RPFs) and other RPFs appearing under soft-edge scaling. The quasi-Gibbs property of RPFs is one of the key ingredients to solve the associated infinite-dimensional stochastic differential equation (ISDE). Because of the divergence of the free potentials and the interactions of the finite particle approximation under soft-edge scaling, the result of the previous paper excludes the Airy RPFs, although Airy RPFs are the most significant RPFs appearing in random matrix theory. We will use the result of the present paper to solve the ISDE for which the unlabeled equilibrium state is the Airy β RPF with β=1,2,4..
6. Hirofumi Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials, Annals of Probability, 10.1214/11-AOP736, 41, 1, 1-49, 2013.01, We investigate the construction of diffusions consisting of infinitely numerous Brownian particles moving in Rd and interacting via logarithmic functions (two-dimensional Coulomb potentials). These potentials are very strong and act over a long range in nature. The associated equilibrium states are no longer Gibbs measures. We present general results for the construction of such diffusions and, as applications thereof, construct two typical interacting Brownian motions with logarithmic interaction potentials, namely the Dyson model in infinite dimensions and Ginibre interacting Brownian motions. The former is a particle system in R, while the latter is in R2. Both models are translation and rotation invariant in space, and as such, are prototypes of dimensions d = 1, 2, respectively. The equilibrium states of the former diffusion model are determinantal or Pfaffian random point fields with sine kernels. They appear in the thermodynamical limits of the spectrum of the ensembles of Gaussian random matrices such as GOE, GUE and GSE. The equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non-Hermitian Gaussian random matrices known as the Ginibre ensemble..
7. Hirofumi Osada, Infinite-dimensional stochastic differential equations related to random matrices, Probability Theory and Related Fields, 10.1007/s00440-011-0352-9, 153, 3-4, 471-509, 2012.08, We solve infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles interacting via two-dimensional Coulomb potentials. The equilibrium states of the associated unlabeled stochastic dynamics are the Ginibre random point field and Dyson's measures, which appear in random matrix theory. To solve the ISDEs we establish an integration by parts formula for these measures. Because the long-range effect of two-dimensional Coulomb potentials is quite strong, the properties of Brownian particles interacting with two-dimensional Coulomb potentials are remarkably different from those of Brownian particles interacting with Ruelle's class interaction potentials. As an example, we prove that the interacting Brownian particles associated with the Ginibre random point field satisfy plural ISDEs..
8. Hirofumi Osada, Tagged particle processes and their non-explosion criteria, Journal of the mathematical society of Japan, 10.2969/jmsj/06230867, 62, 3, 867-894, 2010.06.
Presentations
 Show All Presentations >>
1. Hirofumi Osada, Diffusion in Coulomb environment and a phase transition, Japanese-German Open Conference on Stochastic Analysis 2019, 2019.09.
2. Hirofumi Osada, Random matrices and dynamics of infinite particle systems, Interactions between commutative and non-commutative probability, 2019.08.
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
Educational Activities
In addition to the lectures on the general education, and engineering, and the mathematical science, I am a superviser of sutudents in docter course and master course. I am also concerned with a seminar class of the 3th grade.


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