Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T21:07:22.941Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for a diffusion process with a one-sided Brownian potential

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Kiyoshi Kawazu  and
Yuki Suzuki
Kiyoshi Kawazu*
Affiliation:
Yamaguchi University
Yuki Suzuki*
Affiliation:
Keio University
*
∗Postal address: Department of Mathematics, Faculty of Education, Yamaguchi University, Yoshida, Yamaguchi, 753-8513, Japan.
∗∗Postal address: School of Medicine, Keio University, Hiyoshi, Kouhoku-ku, Yokohama, 223-8521, Japan. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a diffusion process X(t) with a one-sided Brownian potential starting from the origin. The limiting behavior of the process as time goes to infinity is studied. For each t > 0, the sample space describing the random potential is divided into two parts, Ãt and B̃t, both having probability ½, in such a way that our diffusion process X(t) exhibits quite different limiting behavior depending on whether it is conditioned on Ãt or on B̃t (t → ∞). The asymptotic behavior of the maximum process of X(t) is also investigated. Our results improve those of Kawazu, Suzuki, and Tanaka (2001).

Keywords

Random environmentdiffusion processoccupation time

MSC classification

Primary: 60J60: Diffusion processes 60K37: Processes in random environments
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 997 - 1012
Copyright
© Applied Probability Trust 2006 

References

Brox, T. (1986). A one-dimensional diffusion process in a Wiener medium. Ann. Prob. 14, 1206–1218.Google Scholar
Hu, Y. and Shi, Z. (1998). The local time of simple random walk in random environment. J. Theoret. Prob. 11, 765–793.Google Scholar
Itô, K. and McKean, H. P. Jr. (1965). Diffusion Processes and their Sample Paths. Springer, Berlin.Google Scholar
Kawazu, K., Suzuki, Y. and Tanaka, H. (2001). A diffusion process with a one-sided Brownian potential. Tokyo J. Math. 24, 211–229.Google Scholar
Schumacher, S. (1985). Diffusions with random coefficients. In Particle Systems, Random Media and Large Deviations, ed. Durrett, R., American Mathematics Society, Providence, RI, pp. 351–356.Google Scholar
Sinai, Ya. G. (1982). The limiting behavior of a one-dimensional random walk in a random medium. Theory Prob. Appl. 27, 256–268.Google Scholar