Additive functional of one-dimensional diffusion processes

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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.