Stochastic Analysis and its application to differential operators

随机分析及其在微分算子中的应用

• 批准号: 12640117
• 负责人: MATUMOTO Hiroyuki
• 金额: $ 2.24万
• 依托单位: NAGOYA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640117/
• 关键词: Brownian motion; Wiener functionals; Mathematical finance; Hyperbolic spaces; Pitman's theorem; Levy's theorem; Laplacian; グリーン関数; 双曲空間; ウィナー汎関数; ベッセル過程; ブラウン運動; メタプレクティック表現; 対称空間

If we consider a standard Brownian motion and its maximum process, the difference has a same probability law as reflecting Brownian motion and the difference between twice the maximun process and the original Brownian motion has the same law as a three dimensional Bessel process. These results are well known as Levy's and Pitman's theorems. The head investigator Matsumoto, in joint work with Yor, showed analogus or extensions of these theorems for an exponential Brownian functionals by using the classical Laplace method. The exponential Brownian functional appears also in the studies of mathematical finance and in stochastic analysis on hyperbolic spaces. Various aspects of this functional and their applications have been given under the support of this grant. Among others, some explicit expressions for the heat kernels of the semigroups generated by the Laplacians on non-compact symmetric spaces of rank one have been shown by using the results on the exponential Wiener functionals.Other investigators gave many suggestions and comments on the works mentioned above and also worked on their own fields. Ihara studied information theory by applying the theory of large deviations. Uemura studied on the local time of multi-dimensional Brownian motions, which should be considered as generalized Wiener functionals. Ueki studied on the spectral properties of Schrodinger operators with random electro-magnetic fields.

如果我们考虑标准布朗运动及其最大值过程，则其差值具有与反映布朗运动相同的概率规律，并且两倍最大值过程与原始布朗运动之间的差值具有与三维贝塞尔过程相同的规律。这些结果被称为利维定理和皮特曼定理。首席研究员 Matsumoto 与 Yor 合作，通过使用经典的拉普拉斯方法，展示了指数布朗泛函的这些定理的类似或扩展。指数布朗泛函也出现在数学金融研究和双曲空间的随机分析中。该功能的各个方面及其应用都是在这笔赠款的支持下给出的。其中，利用指数维纳泛函的结果，给出了由拉普拉斯算子在一阶非紧对称空间上生成的半群热核的一些显式表达式。其他研究者对所提到的工作提出了许多建议和评论上面也都在自己的领域工作过。伊原应用大偏差理论来研究信息论。上村研究了多维布朗运动的局域时间，应该将其视为广义维纳泛函。 Ueki 研究了具有随机电磁场的薛定谔算子的谱特性。

项目成果

H.Matsumoto, S.Taniguchi: "Wiener functional of second order and their Levy measure"Electric Journal of Probability Theory. 7. 1-30 (2002)

H.Matsumoto、S.Taniguchi：“二阶维纳函数及其 Levy 测度”概率论电气杂志。

ALILI,Larbi, MATSUMOTO,Hiroyuki and SHIRAISHI,Tomoyuki: "On a triplet of exponential Brownian functionals"Sem.Prob.XXXV, Lecture Notes in Math.1755. Springer, Berlin. 396-415 (2001)

ALILI、Larbi、MATSUMOTO、Hiroyuki 和 Shiraishi、Tomoyuki：“关于指数布朗泛函的三元组”Sem.Prob.XXXV，Math.1755 讲义。

Larbi Alili: "On a triplet of exponential Brownian functionals"Seminar de Probabilit'es. 35. 396-415 (2001)

Larbi Alili：“关于指数布朗泛函的三元组”概率研讨会。

MATSUMOTO,Hiroyuki and YOR,Marc: "An analogue of Pitman's 2M-X theorem for exponential Brownian functional, Part II : the role of generalize dinverse Gaussian distributions"Nagoya Math.J.. 162. 65-86 (2001)

MATSUMOTO,Hiroyuki 和 YOR,Marc：“指数布朗泛函的 Pitman 2M-X 定理的类似物，第二部分：推广不同高斯分布的作用”Nagoya Math.J.. 162. 65-86 (2001)

NGUYEN,Laurent, MATSUMOTO,Hiroyuki and YOR,Marc, ed.by R.Buckdahn, H.Engelbert and M.Yor, Gordon and Breach: "Subordinates related to the exponential functionals of Brownian bridges and explicit formulae for the semigroup of hyperbolic Brownian motions, i

NGUYEN,Laurent, MATSUMOTO,Hiroyuki 和 YOR,Marc，R.Buckdahn、H.Engelbert 和 M.Yor、Gordon 和 Breach 编辑：“与布朗桥指数泛函相关的下属以及双曲布朗半群的显式公式