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

  • H. Tanaka
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 9)

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).

Keywords

Brownian Motion Stochastic Differential Equation Limit Distribution Random Environment Exit Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • H. Tanaka
    • 1
  1. 1.Department of Mathematics, Faculty of Science and TechnologyKeio UniversityYokohamaJapan

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