Large deviations for a random walk in random environment. (English) Zbl 0820.60054

Greven, Andreas; den Hollander, Frank

Ann. Probab. 22, No. 3, 1381-1428 (1994).

Let \(\omega = (p_ x)_{x \in \mathbb{Z}}\) be a sequence of i.i.d. r.v.s taking values in (0,1). Given \(\omega\), let \((X_ n)_{n\geq 0}\) be a Markov chain with \(X_ 0 = 0\) and \(X_{n + 1} = X_ n + 1\) (resp. \(X_ n - 1\)) with probability \(p_{X_ n}\) (resp. \(1 - p_{X_ n}\)). It is shown that \(X_ n/n\) satisfies \[ \lim_{n \to \infty} {1\over n} \log P_ \omega(X_ n = [\theta_ n n]) = -I(\theta) \qquad \omega\text{-a.s. as } \theta_ n \to \theta \in [-1,1]. \] The “rate function” \(I\) is calculated explicitly in terms of random continued fractions. This leads to a classification and qualitative description of the shape of \(I\). The paper uses ideas from the authors [Probab. Theory Relat. Fields 91, No. 2, 195-249 (1992; Zbl 0744.60079)] and J.-B. Baillon, Ph. Clément and the authors [J. Reine Angew. Math. 454, 181- 217 (1994; Zbl 0814.49033)].

Reviewer: M.Quine (Sydney)

MSC:

60G50	Sums of independent random variables; random walks
60F10	Large deviations
82C44	Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics

Keywords:

random walk; random environment; large deviations

Citations:

Zbl 0744.60079; Zbl 0814.49033