Discrete Time Harness Processes

TL;DR

This paper investigates the invariant measures and fluctuation limits of discrete-time harness processes in one dimension, establishing ergodic measures and demonstrating convergence of height fluctuations to the Edwards-Wilkinson equation.

Contribution

It constructs a key ergodic invariant measure for the increment process and proves the convergence of height fluctuations to the Edwards-Wilkinson equation.

Findings

Existence of a unique ergodic invariant measure for the increment process.

Height fluctuations converge to the Edwards-Wilkinson equation.

Finite-dimensional marginals extend to process-level convergence.

Abstract

We study the invariant measures and fluctuation limits of discrete-time harness processes in one spatial dimension. We construct one essential ergodic (under spatial shifts) invariant measure of the increment process derived from harness process, and all other ergodic invariant measures can be obtained by adding constants. We also show that the weak limit of the one dimensional height fluctuations starting from the increments under several translation-invariant ergodic measures will obey Edwards-Wilkinson equation, and the finite-dimensional marginal convergence can be extended to a process level convergence.