Research on Diffusion Processes in Random Environments

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1. It was proved that the limit process arising in a suitably scaled reflecting diffusion process, which moves in a Brownian environment of [0, ∞), is a time-inhomogeneous self-similar Levy process. Its Levy measure was computed explicitly.2. It was shown that the long-term behavior of diffusion process with a one-sided Brownian potential in (-∞, ∞) exhibits a phenomenon quite different from those (other) models studied previously. In fact, under the usual Brownian scaling the limit process is a reflecting Brownian motion on [0, ∞) with "probability 1/2" and vanishes identically with "the other probability 1/2".3. As a preliminary discussion toward the investigation of a diffusion process in a multi-dimensional Brownian environment. some new results on extrema of a Brownian motion with a multi-dimensional time were obtained.4. Bochner's subordination of time-homogeneous Markov processes was extended, with modification, to a considerably wider class of time-inhomogeneous Markov processes. In connection with this, some explicit formula for a class of multiplicative functionals of (discontinous) Markov processes was also studied.