Abstract
Suppose that the integers are assigned random variables {ω i} (taking values in the unit interval), which serve as an environment. This environment defines a random walk {X n} (called a RWRE) which, when at i, moves one step to the right with probability ω i, and one step to the left with probability 1 − ω i. When the {ω i} sequence is i.i.d., Greven and den Hollander (1994) proved a large deviation principle for X n/n, conditional upon the environment, with deterministic rate function.We consider in this paper large deviations, both conditioned on the environment (quenched) and averaged on the environment (annealed), for the RWRE, in the ergodic environment case. The annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy. We also give a detailed qualitative description of the resulting rate functions. Our techniques differ from those of Greven and den Hollander, and allow us to present also a trajectorial (quenched) large deviation principle.
References
Alili, S.: Asymptotic behaviour for random walks in random environments, J. Appl. Prob. 36, 334–349 (1999)
Chung, K.L.: Markov chains with stationary transition probabilities, Berlin, Springer, 1960
Dembo, A., Peres, Y., Zeitouni, O.: Tail estimates for one-dimensional random walk in random environment, Commun. Math. Phys. 181, 667–683 (1996)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, second edition, Springer, New York (1998)
Dette, H., Fill, J.A., Pitman, J., Studden, W.J., Wall and Siegmund duality relations for birth and death chains with reflecting barrier, J. Theor. Prob. 10, 349–374 (1997)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time III, Comm. Pure Appl. Math. 29, 389–461 (1976)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time IV, Comm. Pure Appl. Math. 36, 183–212 (1983)
Föllmer, H.: Random fields and diffusion processes. In: Ecole d’Eté de Saint Flour XV-XVII, Springer Lect. Notes in Math. 1362, 101–203 (1988)
Gantert, N., Zeitouni, O.: Quenched sub-exponential tail estimates for one-dimensional random walk in random environment. Commun. Math. Phys. 194, 177–190 (1998)
Georgii, H.-O.: Gibbs Measures and Phase Transitions. De Gruyter, Berlin (1988)
Greven, A., den Hollander, F.: Large deviations for a random walk in random environment, Ann. Probab. 22, 1381–1428 (1994)
Hughes, B. D.: Random walks and random environments, Clarendon Press, Oxford (1996)
Jones, W.B., Thron, W.J.: Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley, 1980
Kesten, H., Kozlov, M.V., Spitzer, F.: A limit law for random walk in a random environment, Comp. Math. 30, 145–168 (1975)
Kozlov, M.V.: Random walk in a one-dimensional random medium, Th. Prob. and Appl. 18, 387–388 (1973)
Ledrappier, F.: Quelques propriétés des exposants charactéristiques, Springer, LNM 1097, 305–396 (1984)
Pisztora, A., Povel, T.: Large deviation principle for random walk in a quenched environment in the low speed regime, Ann. Probab. 27, 1389–1413 (1999)
Pisztora, A., Povel, T., Zeitouni, O.: Precise large deviations estimates for one dimensional random walk in random environment, Probab. Theory Related Fields 113, 171–190 (1999)
Révész, P.: Random walk in random and non-random environments, World Scientific, Singapore (1990)
Solomon, F.: Random walks in random environment, Ann. Probab. 3, 1–31 (1975)
Spitzer, F.: Principles of random walk, Springer, Berlin (1976)
Sznitman, A.-S.: Slowdown and neutral pockets for a random walk in a random environment, Probab. Theory Related Fields 115, 287–323 (1999)
Wall, H.S.: Analytic theory of continued fractions, Van Nostrand, 1948
Zerner, M.P.W.: Lyapunov exponents and quenched large deviations for multi-dimensional random walk in random environment, Ann. Probab. 26, 1446–1476 (1998)
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Research partially supported by CNRS, UMR7599. Part of this work was done while visiting the Dept. of Electrical Eng., Technion, Israel.
Research supported by the Swiss National Science foundation under grant 8220-046518, and partially supported by the DFG. Part of this work was done while visiting the Dept. of Electrical Eng., Technion, Israel.
Partially supported by a US-Israel BSF grant and by IHP.
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Comets, F., Gantert, N. & Zeitouni, O. Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab Theory Relat Fields 118, 65–114 (2000). https://doi.org/10.1007/s004400000074
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DOI: https://doi.org/10.1007/s004400000074
Mathematics Subject Classification (2000)
- 60J15
- 60F10
- 82C44
- 60J80
Key words and phrases
- Random walk in random environment
- Large deviations