論文

Long-time behavior of diffusion processes with one-sided random potentials starting from the origin is studied. As random potentials, some strictly stable processes are given just on the negative side in the real line. This model is an extension of the diffusion process with a one-sided Brownian potential studied by Kawazu, Suzuki and Tanaka (Tokyo J. Math. 24, 211–229 2001) and Kawazu and Suzuki (J. Appl. Probab. 43, 997–1012 2006). In this paper, we analyze our model by different methods from theirs. We use the theory concerning the convergence of a sequence of bi-generalized diffusion processes studied by Ogura (J. Math. Soc. Japan 41, 213–242 1989) and Tanaka (Comm. Pure Appl. Math. 47, 755–766 1994). For diffusion processes with one-sided random potentials, the limit theorems introduced by them cannot be used. We improve their limit theorems and apply the improved limit theorem to examining the long-time behavior of our model. As a result, we show that limit distributions exist under the Brownian scaling with some probability, and under a sub-diffusive scaling with the remaining probability.

We investigate the limiting behavior of diffusion processes in Brownian environments on disconnected selfsimilar fractal sets in R. Due to the effect of Brownian environments, the diffusion processes exhibit ultra-slow diffusive behavior, which are called Brox-type diffusions. We show that the limiting distributions are given under suitable scalings determined by selfsimilar fractal sets and measures related to the sets. The scaling properties are different from that of the Brox-type diffusion on R.

In this paper, we investigate Euler–Maruyama approximate solutions of stochastic differential equations (SDEs) with multiple delay functions. Stochastic differential delay equations (SDDEs) are generalizations of SDEs. Solutions of SDDEs are influenced by both the present and past states. Because these solutions may include past information, they are not necessarily Markov processes. This makes representations of solutions complicated; therefore, approximate solutions are practical. We estimate the rate of convergence of approximate solutions of SDDEs to the exact solutions in the Lp-mean for p ≥ 2 and apply the result to obtain confidence interval estimations for the approximate solutions.