Regional Fractional Stochastic Burgers from random interactions

TL;DR

This paper investigates the transition from Ornstein-Uhlenbeck processes to energy solutions of stochastic Burgers equations with fractional operators, using boundary-driven exclusion processes with long jumps and asymmetric rates.

Contribution

It establishes convergence results to stationary solutions of both processes, depending on asymmetry strength, and leverages recent uniqueness proofs for energy solutions.

Findings

Convergence to Ornstein-Uhlenbeck process in symmetric regime

Convergence to stochastic Burgers equation in asymmetric regime

Utilizes fractional operators like regional fractional Laplacian

Abstract

The purpose of this article is to derive the crossover from the Ornstein-Uhlenbeck process to energy solutions of the stochastic Burgers equation with characteristic operators given in terms of fractional operators, such as the regional fractional Laplacian. The approach is to consider a boundary driven exclusion process with long jumps and asymmetric jump rates. Depending on the strength of the asymmetry we prove the convergence to stationary solutions of either the Ornstein-Uhlenbeck equation, or the stochastic Burgers equation. In the later setting, the convergence in some regimes is guaranteed by the recent proof of uniqueness of energy solutions derived in [16].