On microscopic derivation of a fractional stochastic Burgers equation

TL;DR

This paper derives fractional stochastic PDEs from microscopic particle systems with long-range jumps, showing how different regimes lead to fractional heat or Burgers equations depending on parameters.

Contribution

It provides a microscopic derivation of fractional stochastic heat and Burgers equations from asymmetric particle systems with long-range interactions.

Findings

For lpha<3/2, the fluctuation field converges to a fractional stochastic heat equation.

For lphaer 3/2, the limit satisfies a fractional stochastic Burgers equation.

The type of limiting equation depends on the jump rate structure and asymmetry scale.

Abstract

We derive from a class of microscopic asymmetric interacting particle systems on ${\mathbb Z}$, with long range jump rates of order $|\cdot|^{-(1+\alpha)}$ for $0<\alpha<2$, different continuum fractional SPDEs. More specifically, we show the equilibrium fluctuations of the hydrodynamics mass density field of zero-range processes, depending on the stucture of the asymmetry, and whether the field is translated with process characteristics velocity, is governed in various senses by types of fractional stochastic heat or Burgers equations. The main result: Suppose the jump rate is such that its symmetrization is long range but its (weak) asymmetry is nearest-neighbor. Then, when $\alpha<3/2$, the fluctuation field in space-time scale $1/\alpha:1$, translated with process characteristic velocity, irrespective of the strength of the asymmetry, converges to a fractional stochastic heat equation, the limit also for the symmetric process. However, when $\alpha\geq 3/2$ and the strength of the weak asymmetry is tuned in scale $1-3/2\alpha$, the associated limit points satisfy a martingale formulation of a fractional stochastic Burgers equation.