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June 2018 A Diffusion Process with a Random Potential Consisting of Two Contracted Self-Similar Processes
Yuki SUZUKI
Tokyo J. Math. 41(1): 21-50 (June 2018). DOI: 10.3836/tjm/1502179248

Abstract

We study a limiting behavior of a one-dimensional diffusion process with a random potential. The potential consists of two independent contracted self-similar processes with different indices for the right and the left hand sides of the origin. Brox (1986) and Schumacher (1985) studied a diffusion process with a Brownian potential, and showed, roughly speaking, after a long time with high probability the process is at the bottom of a valley. Their result was extended to a diffusion process in an asymptotically self-similar random environment by Kawazu, Tamura and Tanaka (1989). Our model is a variant of their models. But we show, roughly speaking, after a long time it is possible that our process is not at the bottom of a valley. We also study asymptotic behaviors of the minimum process and the maximum process of our process.

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Yuki SUZUKI. "A Diffusion Process with a Random Potential Consisting of Two Contracted Self-Similar Processes." Tokyo J. Math. 41 (1) 21 - 50, June 2018. https://doi.org/10.3836/tjm/1502179248

Information

Published: June 2018
First available in Project Euclid: 20 November 2017

zbMATH: 06966857
MathSciNet: MR3830807
Digital Object Identifier: 10.3836/tjm/1502179248

Subjects:
Primary: 60J60
Secondary: 60K37

Rights: Copyright © 2018 Publication Committee for the Tokyo Journal of Mathematics

JOURNAL ARTICLE
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Vol.41 • No. 1 • June 2018
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