Quenched large deviations for random walk in a random environment

Yilmaz, Atilla

doi:10.1002/cpa.20283

Cited by 31 publications

(41 citation statements)

References 23 publications

(39 reference statements)

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“…It induces a probability measureP β,θ,ω 0 on nearest-neighbor paths starting at 0.Ê β,θ,ω 0 denotes expectation underP β,θ,ω 0 . With this notation, it is clear from [29,Theorem 5.17]) and we define…”

Section: Variational Formulas For the Tilted Free Energymentioning

confidence: 99%

“…There are alternative proofs of Theorem A.1. One of the authors established in [29] a so-called level-2 LDP from the point of view of the particle performing nearest-neighbor random walk in random environment (RWRE) on Z d , from which the existence of the limit in (A.1) follows as a corollary by Varadhan's integral lemma. That paper built upon the Ph.D. thesis of Rosenbluth [25] who in turn adapted the work of Kosygina, Rezakhanlou and Varadhan [18] on the homogenization of second-order HJ stochastic PDEs with convex Hamiltonians.…”

Section: Appendicesmentioning

confidence: 99%

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Nonconvex homogenization for one-dimensional controlled random walks in random potential

Yilmaz¹,

Zeitouni²

2019

Ann. Appl. Probab.

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We consider a finite horizon stochastic optimal control problem for nearest-neighbor random walk {X i } on the set of integers. The cost function is the expectation of exponential of the path sum of a random stationary and ergodic bounded potential plus θXn. The random walk policies are measurable with respect to the random potential, and are adapted, with their drifts uniformly bounded in magnitude by a parameter δ ∈ [0, 1]. Under natural conditions on the potential, we prove that the normalized logarithm of the optimal cost function converges. The proof is constructive in the sense that we identify asymptotically optimal policies given the value of the parameter δ, as well as the law of the potential. It relies on correctors from large deviation theory as opposed to arguments based on subadditivity which do not seem to work except when δ = 0.The Bellman equation associated to this control problem is a second-order Hamilton-Jacobi (HJ) stochastic partial difference equation with a separable random Hamiltonian which is nonconvex in θ unless δ = 0. We prove that this equation homogenizes under linear initial data to a first-order HJ deterministic partial differential equation. When δ = 0, the effective Hamiltonian is the tilted free energy of random walk in random potential and it is convex in θ. In contrast, when δ = 1, the effective Hamiltonian is piecewise linear and nonconvex in θ. Finally, when δ ∈ (0, 1), the effective Hamiltonian is expressed completely in terms of the tilted free energy for the δ = 0 case and its convexity/nonconvexity in θ is characterized by a simple inequality involving δ and the magnitude of the potential, thereby marking two qualitatively distinct control regimes.

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Section: Variational Formulas For the Tilted Free Energymentioning

confidence: 99%

Section: Appendicesmentioning

confidence: 99%

Nonconvex homogenization for one-dimensional controlled random walks in random potential

Yilmaz¹,

Zeitouni²

2019

Ann. Appl. Probab.

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“…Every limit theorem about this so-called environment Markov chain implies a corresponding limit theorem for the walk. This general and robust approach was first introduced in the context of interacting particle systems [27] and was later successfully adapted to RWRE (see for example [31,34,45]). In light of the previous paragraph, the large deviation behavior of RWDRE can be analyzed via various statistics of either the walk itself or the environment Markov chain.…”

Section: Large Deviation Principles the Point Of View Of The Particlmentioning

confidence: 99%

“…Proof. The proof uses a strategy involving a change-of-measure, Jensen's inequality, and the ergodic theorem, which is standard for obtaining LDP lower bounds for Markov chains, and has been successfully carried out in the context of (undirected) RWRE (see [41,45,35] for the level-1,2,3 quenched LDPs). In fact, keeping future applications in mind, the level-3 quenched LDP lower bound was derived in [35,Section 4] in full detail and without using the assumption that the walk is undirected.…”

Section: Modified Variational Formulas For the Quenched Rate Functionsmentioning

confidence: 99%

Averaged vs. quenched large deviations and entropy for random walk in a dynamic random environment

Rassoul-Agha¹,

Seppäläinen²,

Yilmaz³

2017

Electron. J. Probab.

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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. F P-ess sup ω log z p(z)e V (ω,z)+F (ω,z)

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“…Intensive work to extend both the class of random walks and the class of potentials for which a LDP holds was undertaken by Rassoul-Agha, Seppäläinen, Yilmaz. Some of these results, which include level-3 LDP can be found in [27], [41], [42], [28], [30], [29]. Yilmaz and Zeitouni [43] studied a class of random walks in a random environment where the annealed and quenched rate functions differ.…”

Section: Introductionmentioning

confidence: 99%

Large deviations for Brownian motion in a random potential

Boivin

Lê

2020

ESAIM: PS

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A quenched large deviation principle for Brownian motion in a non-negative, stationary potential is proved. A sufficient moment condition on the potential is given but unlike the results of Armstrong and Tran (2014) no regularity is assumed. The proof is based on a method developed by Sznitman (1994) for Brownian motion among Poissonian potential. In particular, the LDP holds for potentials with polynomially decaying correlations such as the classical potentials studied by L. Pastur (1977) and R. Fukushima (2008) and the potentials recently introduced by H. Lacoin (2012).Subject Classification: 82B41, 60K37

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Quenched large deviations for random walk in a random environment

Cited by 31 publications

References 23 publications

Nonconvex homogenization for one-dimensional controlled random walks in random potential

Nonconvex homogenization for one-dimensional controlled random walks in random potential

Averaged vs. quenched large deviations and entropy for random walk in a dynamic random environment

Large deviations for Brownian motion in a random potential

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