# Fluctuations in 1D stochastic homogenization of pseudo-elliptic equations with long-range dependent potentials

## TL;DR

This paper investigates the effects of long-range dependent random potentials on the homogenization error in 1D pseudo-elliptic equations, showing convergence to Hermite process-based stochastic integrals.

## Contribution

It extends homogenization analysis to long-range dependent potentials, demonstrating convergence to Hermite processes, unlike the short-range case with Brownian motion.

## Key findings

- Homogenization error converges to Hermite process-based stochastic integrals.
- Long-range dependence alters the limit distribution of fluctuations.
- Results differ from short-range dependence scenarios.

## Abstract

This paper deals with the homogenization problem of one-dimensional pseudo-elliptic equations with a rapidly varying random potential. The main purpose is to characterize the homogenization error (random fluctuations), i.e., the difference between the random solution and the homogenized solution, which strongly depends on the autocovariance property of the underlying random potential. It is well known that when the random potential has short-range dependence, the rescaled homogenization error converges in distribution to a stochastic integral with respect to standard Brownian motion. Here, we are interested in potentials with long-range dependence and we prove convergence to stochastic integrals with respect to Hermite process.