[Paper Review] An elliptic Harnack inequality for random walk in balanced environments

Noam Berger, M. Cohen+2 more|arXiv (Cornell University)|Jul 10, 2018

Advanced Mathematical Modeling in Engineering11 references4 citations

TL;DR

This paper establishes an elliptic Harnack inequality for random walks in i.i.d. balanced random environments on $\mathbb{Z}^d$, extending methods from the elliptic case to non-elliptic settings. The key contribution is a quantitative homogenization result showing that discrete harmonic functions converge to their homogenized counterparts with high probability, uniformly on large scales, under a weak moment condition on the environment distribution.

ABSTRACT

We prove a Harnack inequality for the solutions of a difference equation with non-elliptic balanced i.i.d. coefficients. Along the way we prove a (weak) quantitative homogenisation result, which we believe is of some interest too.

Motivation & Objective

• To extend the elliptic Harnack inequality framework—previously developed for elliptic environments—to the non-elliptic case of balanced random environments.
• To establish a quantitative homogenization result for discrete harmonic functions in balanced i.i.d. environments, providing high-probability bounds on the difference between discrete and homogenized solutions.
• To analyze the behavior of random walks in balanced environments where transition probabilities are symmetric but not uniformly bounded away from zero.
• To prove that the quenched invariance principle holds with explicit convergence rates under weak moment conditions on the environment distribution.
• To develop a new method for controlling oscillations of harmonic functions in non-elliptic settings using a recursive geometric argument and environment-dependent scaling.

Proposed method

• Define the environment space $\Omega = (\mathcal{M}^d)^{\mathbb{Z}^d}$ with i.i.d. balanced transition kernels satisfying $\omega(z,e) = \omega(z,-e)$ for all $z \in \mathbb{Z}^d$ and neighbors $e$.
• Introduce the quenched law $P_\omega^z$ and the annealed law $\mathbb{P}^z = \int P_\omega^z \, dP(\omega)$, focusing on the quenched behavior of the walk.
• Use a recursive geometric construction: for a harmonic function $f$, define a sequence of nested balls $B^\text{dis}_{r_j}(z_j)$ and use the Harnack-type inequality to control the ratio $\max f / \min f$ over successive scales.
• Apply a weak moment condition on the environment distribution to ensure that the oscillation of $f$ grows at most polynomially, avoiding exponential blow-up.
• Establish a contradiction by assuming super-exponential growth of $f(x_j)/f(y_j)$, leveraging the fact that such growth violates the Harnack inequality under the given scaling and environment regularity.
• Prove the main result via a contradiction argument based on the oscillation control and the existence of a sequence of points where the ratio $f(x_j)/f(y_j)$ grows faster than allowed by the Harnack principle.

Experimental results

Research questions

• RQ1 Can the elliptic Harnack inequality be extended to non-elliptic balanced random environments?
• RQ2 What is the rate of convergence of discrete harmonic functions to their homogenized counterparts in balanced i.i.d. environments?
• RQ3 How does the absence of uniform ellipticity affect the regularity and oscillation control of harmonic functions in random environments?
• RQ4 Under what moment conditions on the environment distribution does the quenched invariance principle hold with quantitative error bounds?
• RQ5 Can a recursive geometric argument be used to control the ratio of maximum to minimum values of harmonic functions in non-elliptic settings?

Key findings

• The paper proves a Harnack inequality for discrete harmonic functions in balanced i.i.d. random environments, even when the transition probabilities are not uniformly bounded away from zero.
• A quantitative homogenization result is established: the difference between the discrete harmonic function and its homogenized limit is bounded with high probability, uniformly on large scales.
• The oscillation of harmonic functions grows at most polynomially in the scale, preventing super-exponential growth that would contradict the Harnack principle.
• The proof relies on a recursive construction of nested balls and a contradiction argument showing that if the ratio $f(x_j)/f(y_j)$ grows too fast, it violates the Harnack inequality.
• The main result holds under a weak moment condition on the environment distribution, specifically that $\mathbb{E}_Q[\omega(0,e_i)]$ is positive and finite.
• The convergence rate of the quenched invariance principle is quantified via the oscillation control of harmonic functions, with the error decaying polynomially in the scale $R$.