A quenched invariance principle for non-elliptic random walk in i.i.d. balanced random environment

Berger, Noam; Deuschel, Jean–Dominique

doi:10.1007/s00440-012-0478-4

Cited by 43 publications

(109 citation statements)

References 24 publications

(66 reference statements)

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“…One interpretation of the reason for this difference is that some nonlinear equations correspond to stochastic optimal control problems, and the controller is under no obligation to select a stationary control. A variant of our construction leads to a linear counterexample for all p < 1, which was already discovered in [21] (see also [5]) using a similar trap model. The range 1 ≤ p < d thus remains open in the linear case; we believe that p = 1 is the critical exponent.…”

Section: Proposition 32 (Abp Inequality)supporting

confidence: 53%

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

Armstrong¹,

Smart²

2014

Ann. Probab.

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We prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form. The equation is required to be strictly elliptic, but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite dth moment, where d is the dimension. In the general stationary-ergodic framework, we show that the equation homogenizes to a deterministic, uniformly elliptic equation, and we obtain an explicit estimate of the effective ellipticity, which is new even in the uniformly elliptic context. Showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory. We prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity, which are controlled in the macroscopic limit by the ergodic theorem. We show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite pth moment, for every p < d, but for which regularity and homogenization break down. In probabilistic terms, the homogenization results correspond to quenched invariance principles for diffusion processes in random media, including linear diffusions as well as diffusions controlled by one controller or two competing players.

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Section: Proposition 32 (Abp Inequality)supporting

confidence: 53%

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

Armstrong¹,

Smart²

2014

Ann. Probab.

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“…We can use twice Finally, for the sum over the Green's function difference, we recall thatĜ x t coincides with its goodified version. Applying Lemma 3.3 O ((log t) 3…”

Section: Proof Of the Main Technical Statementmentioning

confidence: 96%

An invariance principle for a class of non-ballistic random walks in random environment

Baur

2015

Probab. Theory Relat. Fields

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We are concerned with random walks on Z d , d ≥ 3, in an i.i.d. random environment with transition probabilities ε-close to those of simple random walk. We assume that the environment is balanced in one fixed coordinate direction, and invariant under reflection in the coordinate hyperplanes. The invariance condition was used in [1] as a weaker replacement of isotropy to study exit distributions. We obtain precise results on mean sojourn times in large balls and prove a quenched invariance principle, showing that for almost all environments, the random walk converges under diffusive rescaling to a Brownian motion with a deterministic (diagonal) diffusion matrix. We also give a concrete description of the diffusion matrix. Our work extends the results of Lawler [9], where it is assumed that the environment is balanced in all coordinate directions.

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“…Theorem 1.1 extends the static version of the QCLT proved by Lawler [L82] for uniformly elliptic environments and by Guo and Zeitouni [GZ10] for elliptic environments. Other recent related results for random walks in balanced static environments include Berger and Deuschel [BD14] and Baur [Ba14]. On the other hand, it should be pointed out that several results exist proving QCLT for random walks in time-dependent environments, but in general under mixing condition which are stronger that our ergodicity assumption (see for example [DKL08] or [A14]).…”

Section: Now Setmentioning

confidence: 86%

Quenched invariance principle for random walk in time-dependent balanced random environment

Deuschel¹,

Guo²,

Ramı́rez³

2018

Ann. Inst. H. Poincaré Probab. Statist.

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We prove a quenched central limit theorem for balanced random walks in time-dependent ergodic random environments which is not necessarily nearest-neighbor. We assume that the environment satisfies appropriate ergodicity and ellipticity conditions. The proof is based on the use of a maximum principle for parabolic difference operators. * Electronic address: deuschel@math.tu-berlin.de † Electronic address:

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A quenched invariance principle for non-elliptic random walk in i.i.d. balanced random environment

Cited by 43 publications

References 24 publications

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

An invariance principle for a class of non-ballistic random walks in random environment

Quenched invariance principle for random walk in time-dependent balanced random environment

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