| Project/Area Number |
17K05288
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Tohoku University |
|---|
Principal Investigator |
Hariya Yuu 東北大学, 理学研究科, 教授 (20404030)
|
|---|
| Project Period (FY) |
2017-04-01 – 2022-03-31
|
| Project Status |
Completed (Fiscal Year 2021)
|
| Budget Amount *help |
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
|
|---|
| Keywords | 確率解析 / ブラウン運動 |
|---|
| Outline of Final Research Achievements |
The Ornstein-Uhlenbeck semigroup defined in a Gaussian space is known to possess the hypercontractivity, which property has applications in mathematical physics such as quantum field theory. As one of the achievements of our study, we have applied stochastic analysis to obtain a family of functional inequalities that embraces the above-mentioned hypercontractivity. As for Brownian motion itself on which stochastic analysis is developed, we have revealed some invariance of the law of Brownian motion under a transformation involving its exponential functionals, which is new to our best knowledge, and which we think is of significance in probability theory because of its fundamental feature of Brownian motion.
|
| Academic Significance and Societal Importance of the Research Achievements |
オルンシュタイン＝ウーレンベック半群の超縮小性の発見は，1973年のE. ネルソンによる研究にまでさかのぼるもので，上述の研究成果によりその成立の背後にある一般的な枠組みが明らかになったことは意義あることと思われる．また，ブラウン運動の数学的な構成は1923年にN. ウィナーによりなされ，マルチンゲールやマルコフ過程，またガウス過程といった様々な確率過程のクラスにおける典型例を与えることから，以来確率論においてブラウン運動は基礎的かつ重要な位置を占めてきた．このように現代確率論の中で長い研究の歴史をもつブラウン運動に対して新たな性質が見出されたことは意義深いことと考えられる．
|
