Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariance principles for rough walks in random conductances

Johannes Bäumler, Noam Berger, Tal Orenshtein, Martin Slowik

Abstract

We establish annealed and quenched invariance principles for random walks in random conductances lifted to the $p$-variation rough path topology, allowing for degenerate environments and long-range jumps. The proof is probabilistic and structural: convergence is established by decoupling the martingale lift from terms involving the corrector, specifically the quadratic covariations and the corrector iterated integrals. In the annealed regime, Itô-type techniques ensure that the corrector related terms converge to deterministic processes in the $p$-variation norm in probability. In the quenched regime, reversibility is replaced by the existence of a stationary potential for the corrector with $2+ε$ moments. We also provide a construction of this potential from spatial moment bounds on the corrector and mild volume regularity, which may be of independent interest.

Invariance principles for rough walks in random conductances

Abstract

We establish annealed and quenched invariance principles for random walks in random conductances lifted to the -variation rough path topology, allowing for degenerate environments and long-range jumps. The proof is probabilistic and structural: convergence is established by decoupling the martingale lift from terms involving the corrector, specifically the quadratic covariations and the corrector iterated integrals. In the annealed regime, Itô-type techniques ensure that the corrector related terms converge to deterministic processes in the -variation norm in probability. In the quenched regime, reversibility is replaced by the existence of a stationary potential for the corrector with moments. We also provide a construction of this potential from spatial moment bounds on the corrector and mild volume regularity, which may be of independent interest.
Paper Structure (21 sections, 33 theorems, 294 equations)

This paper contains 21 sections, 33 theorems, 294 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Harmonic embedding and the corrector
  4. Invariance principles for rough paths
  5. Main results: invariance principles for the lifted walk
  6. The annealed invariance principle
  7. The quenched invariance principle
  8. Sources of regular stationary potential: Historical background and a construction
  9. From spatial corrector estimates to stationary potentials
  10. Key lemma in unified settings
  11. The annealed settings - qualitative approach
  12. Harmonic coordinates
  13. Corrector
  14. Corrector p-var vanishes in probability
  15. Tightness of iterated integral of the corrector
  16. ...and 6 more sections

Key Result

Lemma 1.5

For all $\Psi \in L^2_{\mathrm{sol}}$ and $\mathop{\mathrm{\mathbb{P}}}\nolimits$-a.e. $\omega$, we have

Theorems & Definitions (74)

  • Definition 1.2
  • Remark 1.3: Remark about convention of notation
  • Remark 1.4: Remark about extension and restriction
  • Lemma 1.5: Bi11
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Theorem 2.1
  • Remark 2.2: On non-uniqueness of the infinite cluster
  • ...and 64 more