Additive functional of one-dimensional diffusion processes

一维扩散过程的加性泛函

• 批准号: 17540105
• 负责人: KASAHARA Yuji
• 金额: $ 2.22万
• 依托单位: University of Tsukuba
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540105/
• 关键词: diffusion processes; generalized arc-sine law; Brownian motion; random environment; 逆正弦法則; 安定分布; 局所時間; 拡散過程; ベッセル過程; diffusion processes; generalized arc-sine law; Brownian motion; random environment; 逆正弦法則; 安定分布; 局所時間; 拡散過程; ベッセル過程

We studied mainly the long-time asymptotic behavior of additive functionals, especially the occupation times on the positie half line, of one-dimensional diffusion processes. Historically, this problem is well known for Brownian motions and random walks and the limiting distribution obeys the are-sine law. This result has been extended in various ways by many authors. Among them J. Lamperti found the all possible limiting distributions for stochasitic processes with discrete time parameter and he also succeeded to determine the domain of attraction. Although his theorem does not include the case of one-dimensional diffusions, a similar results is shown by S. Watanabe. Many probabilists are still interested in these classical results in connection with financial theory. In our research we studied similar problems for one-dimensional diffusion processes and random walks with random drifts (I. e., in random environments). Our main results are the following: (1) A certain kind of Zero-one law holds. That is, under some technical conditions, the time spent on the positive side converges in distribution to a Bernoulli random variable almost surely. (2) In that case, if the environment is of the stable-type, the time spent on the positive side converges in law to a certain non-degenerate distribution. These results were obtained with S. Watanabe and will be published in Stochastic Processes and its Applications. Another significant result is the following. Y. Yano, et.al. recently proved a functional limit theorem for Lamperti's classical theorem for the occupation times of the positive side. However, they excluded the extreme case of index zero. Our result is that, in such a case, we obtain a functional limit theorem under a non-linear normalization. This result is a joint work with S. Suzuki and published in Proc. Of Japan Acad.

我们主要研究了一维扩散过程的添加功能的长期渐近行为，尤其是在正面线上的职业时间。从历史上看，这个问题以布朗动作和随机步行而闻名，而有限的分布遵守了《明智的法》。许多作者都以各种方式扩展了这一结果。其中J. Lamperti发现了具有离散时间参数的稳态过程的所有可能限制分布，他还成功地确定了吸引人的域。尽管他的定理不包括一维扩散的情况，但沃特那纳布链球菌显示出类似的结果。许多概率主义者仍然对这些与财务理论相关的经典结果感兴趣。在我们的研究中，我们研究了一维扩散过程和随机漂移的随机步行（I. e。，在随机环境中）的类似问题。我们的主要结果是以下内容：（1）某种零一法律所规定。也就是说，在某些技术条件下，几乎可以肯定的是，在分布分布上花费的时间在分布上收敛到伯努利随机变量。 （2）在这种情况下，如果环境是稳定型的，则在正面上花费的时间会在法律上收敛于某些非分数分布。这些结果是用渡边链球菌获得的，将在随机过程及其应用中发表。另一个重要的结果是以下内容。 Y. Yano等最近证明，兰普蒂（Lamperti）的占用时间为正面时代的经典定理是一个功能极限定理。但是，他们排除了指数零的极端情况。我们的结果是，在这种情况下，我们在非线性归一化下获得了功能极限定理。该结果是与S. S. S. S. S. s. s. s. proc。日本学院。