Abstract
We consider random walk with bounded jumps on a hypercubic lattice of arbitrary dimension in a dynamic random environment. The environment is temporally independent and spatially translation invariant. We study the rate functions of the level-3 averaged and quenched large deviation principles from the point of view of the particle. In the averaged case the rate function is a specific relative entropy, while in the quenched case it is a Donsker-Varadhan type relative entropy for Markov processes. We relate these entropies to each other and seek to identify the minimizers of the level-3 to level-1 contractions in both settings. Motivation for this work comes from variational descriptions of the quenched free energy of directed polymer models where the same Markov process entropy appears.
Highlights
After surveying the background of the present work, this introductory section describes the random walk in a dynamic random environment (RWDRE) model and some general notions such as large deviation principles and the point of view of the particle
In the current paper we begin the study of the entropy (1.4). This entropy is the level-2 projection of an entropy that appears in the rate function of a level-3 quenched large deviation principle (LDP) for RWDRE. (See (2.2) and Theorem 2.2 in Section 2.) We study the entropy in this large deviations context
We discover that the connection between the averaged and quenched rate functions can break down rather spectacularly
Summary
After surveying the background of the present work, this introductory section describes the random walk in a dynamic random environment (RWDRE) model and some general notions such as large deviation principles and the point of view of the particle. Minimizing entropy under a mean step condition Eμ[Z1] = ξ as in (1.6) is done in the level-3 to level-1 contraction in large deviation theory For this reason the main focus of the present paper is to study these contractions, both averaged and quenched. I1,a stands for level-1 which is explained two paragraphs below.) Establishing the analogous quenched LDP for (P0ω(Xn/n ∈ · ))n≥1 and identifying the rate function is more arduous It involves considering certain empirical measures from the point of view (POV) of the particle which we introduce next. (i) level-3 averaged LDP for the joint environment-path Markov chain (Section 4); (ii) analysis of the averaged contraction from level-3 to level-1 (Section 5); (iii) alternative formula for the level-3 quenched rate function (Section 6); (iv) relationship of level-3 averaged and quenched rate functions (Section 7); (v) characterizations of the equality of level-1 averaged and quenched rate functions (Section 8); (vi) minimizers of quenched contractions from level-3 to level-1 (Section 9); (vii) spatially constant environments (Section 10)
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