On microscopic derivation of a fractional stochastic Burgers equation

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 $\alpha \ge 3/2$ 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.

24 pages

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Cited by in corpus (3)

• Non-local meta-conformal invariance, diffusion-limited erosion and the XXZ chain
• Spectral Gap Inequality for Long-Range Random Walks
• Non-equilibrium stationary properties of the boundary driven zero-range process with long jumps