Additive functional of one-dimensional diffusion processes

我们主要研究了一维扩散过程的加性泛函的长时间渐近性态，特别是正半直线上的占位时间。在历史上，这个问题对于布朗运动和随机游动是众所周知的，并且极限分布服从正弦定律。这一结果已被许多作者以不同的方式推广。其中J.Lamperti发现了离散时间参数随机过程的所有可能的极限分布，并成功地确定了吸引域。虽然他的定理不包括一维扩散的情况，但S.Watanabe也给出了类似的结果。许多概率学家仍然对这些与金融理论有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具有随机漂移的随机游动(即，在随机环境中)的类似问题。我们的主要结果如下：(1)一类零一定律成立。也就是说，在某些技术条件下，花在正边的时间几乎肯定地按分布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，则在正侧花费的时间在法律上收敛到一定的非退化分布。这些结果是由S.Watanabe获得的，并将发表在《随机过程及其应用》杂志上。另一个重要的结果是。Y.Y.Yano等人最近证明了关于正边占据次数的Lamperti经典定理的一个泛函极限定理。然而，他们排除了指数为零的极端情况。我们的结果是，在这种情况下，我们得到了一个非线性归一化下的泛函极限定理。这一结果是与铃木的合作成果，并发表在Proc上。日本的Acad.

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.