Large deviations for symmetric Markov processes and Dirichlet forms

对称马尔可夫过程和狄利克雷形式的大偏差

• 批准号: 15540103
• 负责人: TAKEDA Masayoshi
• 金额: $ 2.3万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540103/
• 关键词: Markov process; Dirichlet form; Feynman-Kac functional; time-changed process; ファインマンーカッツ汎関数; ファインマン-カッツの公式; 対称マルコフ過程; 大偏差原理; 分枝対称マルコフ過程; 対称安定過程

We proved that the integrability of Feynman-Kac functionals (gaugeability) is equivalent to that the principal eigenvalue of time-changed process is greater than 1, which is also equivalent to the subcriticality of Schroedinger operators. This fact says that the principal eigenvalue of time-changed process accurately measures the size of measures. Using this fact, we obtained three results : The first result is that the ultracontractiyity of Schroedinger semigroups holds if and only if the princilal eigenvalue of time-changed process is greater than 1. The second result is that the expectation of the number of branches hitting a closed set in a branching symmetric stable process is finite if and only if the princilal eigenvalue is greater than 1. The final result is as foolows : Suppose that the heat kernel on a complete Riemannian manifold satisfies the global Gaussian bounds, so called Li-Yau estimate. Then the heat kernel of the Schroedinger operator also possesses the global Gaussian bounds, if and anly if the princilal eigenvalue is greater than 1.

证明了Feynman-Kac泛函的可积性（可测性）等价于时变过程的主特征值大于1，也等价于薛定谔算子的次临界性。这一事实说明时变过程的主特征值准确地度量了度量的大小。利用这一事实，我们得到了三个结果：第一，当且仅当时变过程的主特征值大于1时，薛定谔半群的超收缩性成立；第二个结果是当且仅当主特征值大于1时，分支对称稳定过程中碰到闭集的分支数的期望是有限的。最后的结果如下：假设完全黎曼流形上的热核满足全局高斯边界，即Li-Yau估计。则当且仅当主特征值大于1时，薛定谔算子的热核也具有全局高斯边界。

项目成果

Absolute continuity of symmetric Markov processes

对称马尔可夫过程的绝对连续性

• 发表时间: 2004
• 期刊: Ann.Probab. 32
• 作者: 竹田雅好;服部哲弥;塩谷隆;M.Takeda;T.Hattori;T.Shioya;服部哲弥;塩谷隆;竹田 雅好;竹田雅好;M.Takeda;竹田 雅好
• 通讯作者: 竹田 雅好

Differentiability of spectral functions for symmetric α-stable processes.

对称 α 稳定过程的谱函数的可微性。

• 期刊: Trans. Amer. Math. Soc. (掲載確定)
• 作者: M.Takeda;K.Tsuchida
• 通讯作者: K.Tsuchida

Variational formula for Dirichlet forms and estimates of principal eigenvalues for symmetric α-stable process

对称α稳定过程的狄利克雷形式的变分公式和主特征值的估计

• 发表时间: 2005
• 期刊: Potential Analysis (in press)(to appear)
• 作者: 竹田雅好;服部哲弥;塩谷隆;M.Takeda;T.Hattori;T.Shioya;服部哲弥;塩谷隆;竹田 雅好
• 通讯作者: 竹田 雅好

竹田 雅好: "Subcriticality and Gaugeability for symmetric α-stable processes"Fourm Math.. (to appear).

Masayoshi Takeda：“对称 α 稳定过程的亚临界性和可测性”Fourm Math..（即将出现）。

竹田 雅好: "Criticality of generalized Scbvodinger operators and differentiability of spectral functious"Advanced Studies in Pure Mathematics. 41. 333-350 (2004)

Masayoshi Takeda：“广义 Scbvodinger 算子的临界性和谱函数的可微性”纯数学高级研究 41. 333-350 (2004)。