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Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. (English) Zbl 0965.60098

Probab. Theory Relat. Fields 118, No. 1, 65-114 (2000); erratum ibid. 125, No. 1, 42-44 (2003).
Assume that the integers \(i\in\mathbb{Z}\) are assigned random variables \(\omega_i\) taking values in the unit interval, which serve as an environment. This environment defines a random walk \((X_n)_{n\geq 0}\) (called an RWRE) which, when at \(i\), moves one step to the right with probability \(\omega_i \) and one step to the left with probability \(1-\omega_i\). When \((\omega_i)_{i \in\mathbb{Z}}\) is i.i.d., A. Greven and F. den Hollander [Ann. Probab. 22, No. 3, 1381-1428 (1994; Zbl 0820.60054)] proved a large deviation principle for \(X_n/n\), conditional upon the environment, with deterministic rate function. The present authors consider 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. A detailed qualitative description of the resulting rate functions is also given. The authors’ techniques differ from those of Greven and den Hollander, and allow one also to present a trajectorial (quenched) large deviation principle.

MSC:

60K37 Processes in random environments
60F10 Large deviations
60G50 Sums of independent random variables; random walks

Citations:

Zbl 0820.60054
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References:

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