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Central limit theorem for random walk in degenerate divergence-free random environment: $\mathcal H_{-1}$ reloaded with relaxed ellipticity

Bálint Tóth

Abstract

This paper enhances the result of the work [G. Kozma, B. Tóth, Ann. Probab. vol. 45 (2017) 4307-4347] . We prove the central limit theorem (in probability w.r.t. the environment) for the displacement of a random walker in divergence-free (or, doubly stochastic) random environment, with substantially relaxed ellipticity assumptions. Integrability of the reciprocal of the symmetric part of the jump rates is only assumed (rather than their boundedness, as in previous works on this type of RWRE). Relaxing ellipticity involves substantial changes in the proof, making it conceptually elementary in the sense that it does not rely on Nash's inequality in any disguise.

Central limit theorem for random walk in degenerate divergence-free random environment: $\mathcal H_{-1}$ reloaded with relaxed ellipticity

Abstract

This paper enhances the result of the work [G. Kozma, B. Tóth, Ann. Probab. vol. 45 (2017) 4307-4347] . We prove the central limit theorem (in probability w.r.t. the environment) for the displacement of a random walker in divergence-free (or, doubly stochastic) random environment, with substantially relaxed ellipticity assumptions. Integrability of the reciprocal of the symmetric part of the jump rates is only assumed (rather than their boundedness, as in previous works on this type of RWRE). Relaxing ellipticity involves substantial changes in the proof, making it conceptually elementary in the sense that it does not rely on Nash's inequality in any disguise.
Paper Structure (23 sections, 14 theorems, 124 equations)

This paper contains 23 sections, 14 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integrability conditions on the jump rates
  4. The CLT
  5. Examples
  6. Adding stream term to random conductances
  7. Two totally asymmetric examples
  8. RWRE with stationary density
  9. Related earlier works
  10. Blueprint of the proof and structure of the paper
  11. Spaces and operators
  12. Basic spaces and operators
  13. The infinitesimal generator - acting on $\mathcal{L}_2$
  14. $\mathcal{H}_{+}$, $\mathcal{H}_{-}$, Riesz operators, isometries
  15. Abstract setting
  16. ...and 8 more sections

Key Result

Proposition 1

Assuming r-l2 and h-l1,

Theorems & Definitions (22)

  • Proposition 1: Strong Law of Large Numbers
  • Theorem 2: Main theorem: CLT
  • Proposition 3: The infinitesimal generator
  • Proposition 4: $\mathcal{H}_{\pm}$ spaces, general abstract setting
  • Proposition 5: $\mathcal{H}_{\pm}$ spaces, concrete setting
  • proof : Proof of Proposition \ref{['prop: Hpm particular']}
  • Lemma 6: $\mathcal{H}_-$-bounds on drifts
  • proof
  • Proposition 7: Diffusive upper bound
  • proof
  • ...and 12 more