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Peer-reviewedCC BY 4.0Probability Theory and Related Fields

Superdiffusive central limit theorem for a class of driven diffusive systems at the critical dimension

  • Giuseppe Cannizzaro1
  • Tom Klose2
  • Quentin Moulard3
  1. 1University of Warwick, Coventry, UK
  2. 2University of Oxford, Oxford, UK
  3. 3Technical University of Vienna, Vienna, Austria
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Abstract

We study the large-scale behaviour of a class of driven diffusive systems modelled by a Stochastic Partial Differential Equation, the Stochastic Burgers Equation (SBE) with general nonlinearity, at the critical dimension and in infinite volume. Our main result shows that, under a logarithmically superdiffusive space-time scaling, it is given by the same explicit Gaussian Fixed point obtained in [G. Cannizzaro, Q. Moulard, & F. Toninelli, https://arxiv.org/abs/2501.00344, 2025] for the quadratic SBE, but with suitably renormalised coefficients, thereby rigorously justifying and partly correcting the classical Physics derivation of the SBE in [H. van Beijeren, R. Kutner, & H. Spohn, Phys. Rev. Lett., 1986] based on Spohn’s theory of nonlinear fluctuating hydrodynamics. Besides, ours is the first universality-type result for out-of-equilibrium systems and the first extension of [M. Hairer, J. Quastel, Forum of Mathematics, Pi, Vol. 6, 2018, e3], to the critical dimension and beyond weak coupling. The major challenge in our work is the mild growth condition on the nonlinearity which renders even the well-posedness of the microscopic equation non-trivial. Additional key novelties include the derivation of fine estimates on the non-quadratic part of the generator as well as a new approximation for the resolvent associated to the solution of the quadratic SBE.

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Cite

BibTeXRIS
BibTeX
@article{cannizzaro2026superdiffusive,
  title = {{Superdiffusive central limit theorem for a class of driven diffusive systems at the critical dimension}},
  author = {Giuseppe Cannizzaro and Tom Klose and Quentin Moulard},
  journal = {Probability Theory and Related Fields},
  year = {2026},
  month = jul,
  doi = {10.1007/s00440-026-01515-8},
  url = {https://doi.org/10.1007/s00440-026-01515-8},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Superdiffusive central limit theorem for a class of driven diffusive systems at the critical dimension
AU  - Giuseppe Cannizzaro
AU  - Tom Klose
AU  - Quentin Moulard
PY  - 2026
DA  - 2026/07/15
JO  - Probability Theory and Related Fields
DO  - 10.1007/s00440-026-01515-8
UR  - https://doi.org/10.1007/s00440-026-01515-8
AB  - We study the large-scale behaviour of a class of driven diffusive systems modelled by a Stochastic Partial Differential Equation, the Stochastic Burgers Equation (SBE) with general nonlinearity, at the critical dimension and in infinite volume. Our main result shows that, under a logarithmically superdiffusive space-time scaling, it is given by the same explicit Gaussian Fixed point obtained in [G. Cannizzaro, Q. Moulard, & F. Toninelli, https://arxiv.org/abs/2501.00344, 2025] for the quadratic SBE, but with suitably renormalised coefficients, thereby rigorously justifying and partly correcting the classical Physics derivation of the SBE in [H. van Beijeren, R. Kutner, & H. Spohn, Phys. Rev. Lett., 1986] based on Spohn’s theory of nonlinear fluctuating hydrodynamics. Besides, ours is the first universality-type result for out-of-equilibrium systems and the first extension of [M. Hairer, J. Quastel, Forum of Mathematics, Pi, Vol. 6, 2018, e3], to the critical dimension and beyond weak coupling. The major challenge in our work is the mild growth condition on the nonlinearity which renders even the well-posedness of the microscopic equation non-trivial. Additional key novelties include the derivation of fine estimates on the non-quadratic part of the generator as well as a new approximation for the resolvent associated to the solution of the quadratic SBE.
SN  - 0178-8051
SN  - 1432-2064
ER  -