Occupation time theorems for one-dimensional random walks and diffusion processes in random environments. (English) Zbl 1161.60025
Already at the beginning of the paper the authors develop a general theory for the growth in time, in the log scale, of a class of one-dimensional Brownian additive functionals. These results they apply to obtain asymptotic growth in time of occupation times of a family of one-dimensional diffusion processes. In the next step the authors achieve the long time asymptotics of occupation times for diffusions (and random walks) in random environments. They fundamentally assume that the environment is self-similar in the case of diffusion and asymptotically self-similar in the case of random walks. The environments on the positive and negative sides are independent symmetric stable processes which may have different exponents. The case of random walks is studied by imbedding them in birth-and-death processes with asymptotically self-similar environments.
Reviewer: Elisaveta Pancheva (Sofia)
MSC:
| 60J55 | Local time and additive functionals |
| 60J60 | Diffusion processes |
| 60G51 | Processes with independent increments; Lévy processes |
| 60G52 | Stable stochastic processes |
Keywords:
occupation times; arc-sine law; Lamperti laws; Brox model; random environments; Feller generator; functional limit theoremReferences:
| [1] | Bertoin, J., A ratio ergodic theorem for Brownian additive functionals with infinite mean, Potential Anal., 7, 615-621 (1997) · Zbl 0885.60073 |
| [2] | Blumenthal, R. M.; Getoor, R. K.; Ray, D. B., On the distribution of the first hits for the symmetric stable processes, Trans. Amer. Math. Soc., 99, 540-554 (1961) · Zbl 0118.13005 |
| [3] | Bourbaki, N., Topologie Générale, (Utilisation des Nombre Réels en Topologie Generale (1958), Herman: Herman Paris), (Chapter 9) · Zbl 0031.05502 |
| [4] | Bourbaki, N., Intégration, (Intégration sur les Espaces Topologiques Séparés (1969), Herman: Herman Paris), (Chapter 9) · Zbl 0189.14201 |
| [5] | Brox, T. H., A one-dimensional diffusion process in a Wiener medium, Ann. Probab., 14, 1206-1218 (1986) · Zbl 0608.60072 |
| [6] | Hu, Y.; Zhan, S., The local time of simple random walk in random environment, J. Theoret. Probab., 11, 765-793 (1998) · Zbl 0914.60050 |
| [7] | Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, (North-Holland Mathematical Library, vol. 24 (1989), North-Holland: North-Holland Amsterdam, Kodansha, Tokyo) · Zbl 0205.43702 |
| [8] | Itô, K.; McKean, H. P., Diffusion Processes and Their Sample Paths (1965), Springer, in Classics in Math. 1974 · Zbl 0127.09503 |
| [9] | Kasahara, Y.; Watanabe, S., Occupation time theorems for a class of one-dimensional diffusion processes, Periodica Math. Hungar., 50, 175-188 (2005) · Zbl 1098.60080 |
| [10] | Kasahara, Y.; Watanabe, S., Brownian representation of a class of Lévy processes and its application to occupation times of diffusion processes, Illinois J. Math., 50, 515-539 (2006) · Zbl 1102.60066 |
| [11] | Kawazu, K.; Tamura, Y.; Tanaka, H., One-dimensional diffusions and random walks in random environments, (Watanabe, S.; Prokhorov, Yu. V., Probability Theory and Mathematical Statistics. Probability Theory and Mathematical Statistics, Lecture Notes in Math., vol. 1299 (1988), Springer), 170-184 · Zbl 0637.60091 |
| [12] | Lindvall, T., Weak convergence of probability measures and random functions in the function space \(D [0, \infty)\), J. Appl. Probab., 10, 109-121 (1973) · Zbl 0258.60008 |
| [13] | Parthasarathy, K. R., Probability measures on metric spaces, (Probability and Mathematical Statistics, vol. 3 (1967), Academic Press, Inc.: Academic Press, Inc. New York, London) · Zbl 1188.60001 |
| [14] | Y. Suzuki, A diffusion process with a random potential consisting of two self-similar processes with different indices, preprint; Y. Suzuki, A diffusion process with a random potential consisting of two self-similar processes with different indices, preprint · Zbl 1172.60033 |
| [15] | Watanabe, S., Generalized arc-sine laws for one-dimensional diffusion processes and random walks, (Stochastic Analysis, Ithaca, NY, 1993. Stochastic Analysis, Ithaca, NY, 1993, Proc. Sympos. Pure Math., vol. 57 (1995), AMS), 157-172 · Zbl 0824.60080 |
| [16] | Zeitouni, O., Random walks in random environment, (Lectures on Probability Theory and Statistics. Lectures on Probability Theory and Statistics, Lecture Notes in Math., vol. 1837 (2004), Springer: Springer Berlin), 189-312 · Zbl 1060.60103 |
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