Central limit theorem for random walks in doubly stochastic random environment: ${\mathscr{H}_{-1}}$ suffices

We prove a central limit theorem under diffusive scaling for the displacement of a random walk on ZdZd in stationary and ergodic doubly stochastic random environment, under the H−1H−1-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer], where it is assumed that the stream tensor is in Lmax{2+δ,d}Lmax{2+δ,d}, with δ>0δ>0. Our proof relies on an extension of the relaxed sector condition of [Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012) 463–476], and is technically rather simpler than existing earlier proofs of similar results by Oelschlager [Ann. Probab. 16 (1988) 1084–1126] and Komorowski, Landim and Olla [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer].

• Publication year

  2017

• Venue

  Annals of Probability

• Publication date

  2017-02-22

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1214/16-AOP1166arXiv 1702.06905
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Poisson–Furstenberg boundary and growth of groups

  2016cited by this paper
• Methods Of Modern Mathematical Physics

  2016cited by this paper
• Random walks on disordered media and their scaling limits

  2014cited by this paper
• Central limit theorem for random walks in divergence-free random drift field: "H-minus-one" suffices

  2014cited by this paper
• Lecture Notes on Random Walks in Random Environments

  2013cited by this paper
• Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation

  2012influential reference
• Relaxed sector condition

  2012influential reference
• Poisson–Furstenberg boundary and growth of groups

  2011cited by this paper
• Recent progress on the Random Conductance Model

  2011cited by this paper
• Entropy and the Quantum

  2010cited by this paper
• Superdiffusive Bounds on Self-repellent Brownian Polymers and Diffusion in the Curl of the Gaussian Free Field in d=2

  2010cited by this paper
• High-Dimensional Lipschitz Functions are Typically Flat

  2010influential reference
• Diffusivity bounds for 1D Brownian polymers

  2009cited by this paper
• TRACE INEQUALITIES AND QUANTUM ENTROPY: An introductory course

  2009cited by this paper
• Quenched invariance principle for simple random walk on percolation clusters

  2005cited by this paper
• Evolving sets, mixing and heat kernel bounds

  2003influential reference
• On the sector condition and homogenization of diffusions with a Gaussian drift

  2003influential reference
• A note on the central limit theorem for two-fold stochastic random walks in a random environment

  2003influential reference
• ISOPERIMETRIC INEQUALITIES: DIFFERENTIAL GEOMETRIC AND ANALYTIC PERSPECTIVES (Cambridge Tracts in Mathematics 145) By ISAAC CHAVEL: 268 pp., £50.00, ISBN 0-521-80267-9 (Cambridge University Press, 2001).

  2002cited by this paper
• On the Superdiffusive Behavior of Passive Tracer with a Gaussian Drift

  2002influential reference
• Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives

  2001cited by this paper
• Central limit theorems for tagged particles and for diffusions in random environment

  2001influential reference
• Diffusive limit of a tagged particle in asymmetric simple exclusion processes

  2000cited by this paper
• Self Diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion

  1995influential reference
• Homogenization of a Diffusion Process in a Divergence-Free Random Field

  1988influential reference
• Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions

  1986cited by this paper
• Persistent random walks in random environment

  1986influential reference
• The method of averaging and walks in inhomogeneous environments

  1985influential reference
• Homogenization of diffusion processes with random stationary coefficients

  1983cited by this paper
• Central Limit Theorems for Martingales with Discrete or Continuous Time

  1982cited by this paper
• An Invitation To C*-Algebras

  1976cited by this paper
• Über monotone Matrixfunktionen

  1934cited by this paper

• The mean square displacement of random walk on the Manhattan lattice

  2022cites this paper
• Variable speed symmetric random walk driven by the simple symmetric exclusion process

  2022cites this paper
• logt-Superdiffusivity for a Brownian particle in the curl of the 2D GFF

  2021cites this paper
• Variable speed symmetric random walk driven by symmetric exclusion

  2021cites this paper
• Large-scale regularity in stochastic homogenization with divergence-free drift

  2020cites this paper
• Hydrodynamic limit of the zero range process on a randomly oriented graph

  2020cites this paper
• Anisotropic bootstrap percolation in three dimensions

  2019cites this paper
• Stochastic Simulations of Diffusion Processes

  2019cites this paper
• Polynomial ballisticity conditions and invariance principle for random walks in strong mixing environments

  2019cites this paper
• Deterministic walks in random environment

  2018cites this paper
• DIFFUSIVE AND SUPER-DIFFUSIVE LIMITS FOR RANDOM WALKS AND DIFFUSIONS WITH LONG MEMORY

  2018cites this paper
• Analysis of random walks in dynamic random environments viaL2-perturbations

  2016cites this paper