Limit Distributions for One-Dimensional Diffusion Processes in Self-Similar Random Environments

Abstract

Let X(t) be the one-dimensional diffusion process described by the stochastic differential equation $$ \text{dx(t) = dB(t) - }\frac{1}{2}{{\text{W}}^{1}}(\text{x}(\text{t}))\text{dt, x(0) = 0,} $$ (1) where B(t) is a one-dimensional Brownian motion starting at 0 and {W(x), x∈IR} is a random environment which is independent of the Brownian motion B(t). We are interested in the asymptotic behavior of X(t) as t → ∞: Under what scaling does X(t) have a limit distribution? Similar problems for random walks were considered by Kesten, Kozlov and Spitzer [5] and Sinai [8]. The problem we discuss here is a diffusion analogue of Sinai’s random walk problem [8]. In the case of a Brownian environment Brox [1] proved that the distribution of (log t)−2X(t) is convergent as t → ∞. Similar results were obtained by Schumacher [7] for a considerably wider class of self-similar random environments. As was seen by these works the assumption of the self-similarity of the random environment is important and the notion of suitably defined valleys of the environment plays a central role in the proof. However, it was assumed that the environment has only one point which gives the same value of local minima or maxima (the bottom of a valley consists of a single point), and the explicit form of the limit distribution was unknown until a recent discovery by Kesten ([6]) for Sinai’s random walk which corresponds to the case of a Brownian environment in our diffusion setup (Golosov also obtained the same result as Kesten’s; see also Golosov [2] for the corresponding result in another different model).