An invariance principle for reversible Markov processes. Applications to random motions in random environments

Masi, Anna De; Ferrari, Pablo A.; Wick, W. David

doi:10.1007/bf01041608

Cited by 271 publications

(363 citation statements)

References 30 publications

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“…(The results of [13,14] were primarily two-dimensional but, with the help of [3], they apply to all d ≥ 2; cf [38].) A number of proofs of quenched invariance principles have appeared in recent years for the cases where an annealed principle was already known.…”

Section: Discussion and Related Workmentioning

confidence: 99%

Quenched invariance principle for simple random walk on percolation clusters

Berger

Biskup²

2006

Probab. Theory Relat. Fields

191

258

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Abstract. We consider the simple random walk on the (unique) infinite cluster of supercritical bond percolation in Z d with d ≥ 2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.

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Section: Discussion and Related Workmentioning

confidence: 99%

Quenched invariance principle for simple random walk on percolation clusters

Berger

Biskup²

2006

Probab. Theory Relat. Fields

191

258

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“…In this way, we extend the non-localisation theorem of [21] in order to obtain a central limit theorem. This last theorem may be viewed as a generalisation of certain results in [11,22].…”

Section: Introductionmentioning

confidence: 52%

“…Our basic strategy in proving the central limit theorem is to adapt the arguments proposed by Kipnis and Varadhan [24] and further developed by DeMasi, Ferrari, Goldstein, and Wick [10,11]. Substantial difficulties arise in following this strategy.…”

Section: Introductionmentioning

confidence: 99%

A central limit theorem for random walks in random labyrinths

Bezuidenhout

Grimmett

1999

Annales De l'Institut Henri Poincare (B) Probability and Statistics

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Abstract. A beam of light shines through the lattice Z d , and is subjected to reflections determined by a random environment of mirrors at the vertices of Z d . The behaviour of the light ray is investigated under the hypothesis that the environment contains a strictly positive density of vertices at which the light behaves in the manner of a random walk. When d ≥ 2 and the density of non-trivial reflectors is sufficiently small, the environment contains a.s. a unique infinite 'inter-illuminating' class of vertices. Furthermore, when the light beam originates within this class, then its trajectory obeys a functional central limit theorem with a strictly positive diffusion constant. These facts are obtained using percolation-type arguments, together with the invariance principle proposed by Kipnis and Varadhan.

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“…It is well known ([KV86], [MFGW89], [CI01]) that in the annealed regime (under the law µ ⊗ P ξ(η) 0 ) as ǫ ↓ 0, ǫX t/ǫ 2 , ξ(η) converges in law towards a Brownian Motion with covariance matrix (effective diffusivity) D(ξ, µ).…”

Section: The Ergodic Homogenization Problemmentioning

confidence: 99%

Approximation of the effective conductivity of ergodic media by periodization

Owhadi¹

2003

Probab Theory Relat Fields

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This paper is concerned with the approximation of the effective conductivity σ(A, µ) associated to an elliptic operator ∇ x A(x, η)∇ x where foris a bounded elliptic random symmetric d × d matrix and η takes value in an ergodic probability space (X, µ).We extend this result to non-symmetric operators ∇ x (a + E(x, η))∇ x corresponding to diffusions in ergodic divergence free flows (a is d × d elliptic symmetric matrix and E(x, η) an ergodic skew-symmetric matrix); and to discrete operators corresponding to random walks on Z d with ergodic jump rates. The core of our result is to show that the ergodic Weyl decomposition associated to L 2 (X, µ) can almost surely be approximated by periodic Weyl decompositions with increasing periods, implying that semi-continuous variational formulae associated to L 2 (X, µ) can almost surely be approximated by variational formulae minimizing on periodic potential and solenoidal functions.

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An invariance principle for reversible Markov processes. Applications to random motions in random environments

Cited by 271 publications

References 30 publications

Quenched invariance principle for simple random walk on percolation clusters

Quenched invariance principle for simple random walk on percolation clusters

A central limit theorem for random walks in random labyrinths

Approximation of the effective conductivity of ergodic media by periodization

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