2020 Fiscal Year Final Research Report
Study on diffusion processes by analytical methods
| Project/Area Number |
18K03342
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Review Section |
Basic Section 12010:Basic analysis-related
|
|---|
| Research Institution | Aoyama Gakuin University |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2018-04-01 – 2021-03-31
|
| Keywords | 拡散過程 / 生成作用素 / ラプラス-ベルトラミ作用素 / ブラウン運動 |
|---|
| Outline of Final Research Achievements |
I showed that the Laplace-Beltrami operator on the space of positive definite matrices is represented in a simple way by using the Iwasawa coordinate and that the corresponding diffusion process, called the Brownian motion, may be expressed explicitly as a Wiener functional. Moreover, it is shown that the determinant process is a geometric Brownian motion. We also studied the long time asymptotic behavior of the largest and the smallest eigenvalue processes and showed a central limit theorem, if we regard the corresponding known results as a law of large numbers. Part of this work is a joint one with a Master course student, Jun Otani.
Some years ago, I proved that two dimensional diffusion processes which are generated by the Laplace-Beltrami operators are obtained by time change of usual two-dimensional Brownian motion under additional assumptions. I tried to go further, but I could not obtained a better result.
|
| Free Research Field |
確率論
|
| Academic Significance and Societal Importance of the Research Achievements |
正定値行列は多変量解析の分野では共分散行列として現れる．また，この行列全体は対称空間，等質空間の重要な例として古くから研究されてきた．このような様々な方向から考察される空間においてラプラス-ベルトラミ作用素の簡明な形を与えることは，セルバーグ跡公式などへの応用が考えられる重要な結果だと考えている．
一次元拡散過程が常に一次元ブラウン運動の時間変更で表現され，三次元以上の場合は生成作用素がラプラス-ベルトラミ作用素でもブラウン運動の時間変更では書けないことから，二次元拡散過程はどちらとも異なる重要な研究対象である．等温座標の存在に深く関係するという意味でも重要であると思われる．
|
