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

Giuseppe Cannizzaro, Tom Klose, Quentin Moulard

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, 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.

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

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, 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.
Paper Structure (18 sections, 24 theorems, 88 equations)

This paper contains 18 sections, 24 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Physical motivation and background
  3. The model and main results
  4. Discussion of the main results
  5. Comparison to the literature
  6. Preliminaries
  7. Notations, Function spaces and Wiener space analysis
  8. Gaussian white noise and Fock spaces
  9. Analysis of the microscopic equation
  10. Well-posedness for the microscopic equation
  11. The semigroup and the generator of the generalised SBE
  12. Proof of the main result
  13. Main ideas and roadmap
  14. Estimates for the non-quadratic part of the generator
  15. The amended Ansatz for the resolvent of the quadratic Burgers equation
  16. ...and 3 more sections

Key Result

Proposition 1.2

Let $F$ be such that Assumption Assumption:F holds. Then, e:GeneralisedSBE admits a unique global-in-time solution $u$. Further, $u$ is Markov and has $\mathbb{P}^\varrho$ as invariant measure, where $\mathbb{P}^\varrho$ is the law of $\eta^1\stackrel{\hbox{\tiny def}}{=} \eta\ast\varrho$ for $\eta$

Theorems & Definitions (38)

  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • ...and 28 more