TY - JOUR

T1 - On Microscopic Derivation of a Fractional Stochastic Burgers Equation

AU - Sethuraman, Sunder

N1 - Funding Information:
We would like to thank the referees for their careful reading of a previous version of the manuscript, and their constructive criticism. Thanks also to Tadahisa Funaki and Jeremy Quastel for discussions on the fractional KPZ Burgers equation (1.4). This work was supported in part by ARO Grant W911NF-14-1-0179.
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - We derive, from a class of asymmetric mass-conservative interacting particle systems on Z, with long-range jump rates of order | · |−(1+α) for 0 < α < 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 structure of the asymmetry, and whether the field is translated with process characteristics velocity, are 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 α < 3/2, the fluctuation field in space-time scale 1/α : 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 α ≥ 3/2 and the strength of the weak asymmetry is tuned in scale 1 – 3/2α, the associated limit points satisfy a martingale formulation of a fractional stochastic Burgers equation.

AB - We derive, from a class of asymmetric mass-conservative interacting particle systems on Z, with long-range jump rates of order | · |−(1+α) for 0 < α < 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 structure of the asymmetry, and whether the field is translated with process characteristics velocity, are 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 α < 3/2, the fluctuation field in space-time scale 1/α : 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 α ≥ 3/2 and the strength of the weak asymmetry is tuned in scale 1 – 3/2α, the associated limit points satisfy a martingale formulation of a fractional stochastic Burgers equation.

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U2 - 10.1007/s00220-015-2524-4

DO - 10.1007/s00220-015-2524-4

M3 - Article

AN - SCOPUS:84953368970

SN - 0010-3616

VL - 341

SP - 625

EP - 665

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -