(log t)²/3-superdiffusivity for the 2d stochastic Burgers equation

The Stochastic Burgers equation was introduced in [H. van Beijeren, R. Kutner and H. Spohn, Excess noise for driven diffusive systems, PRL, 1985] as a continuous approximation of the fluctuations of the asymmetric simple exclusion process. It is formally given by ∂ₜη =1/2Δη+ w·∇(η²) + ∇·ξ, where ξ is d-dimensional space time white noise and w is a fixed non-zero vector. In the critical dimension $d=2$ at stationarity, we show that this system exhibits superdiffusve behaviour: more specifically, its bulk diffusion coefficient behaves like (log t)²3, in a Tauberian sense, up to logloglog t corrections. This confirms a prediction made in the physics literature and complements [G. Cannizzarro, M. Gubinelli, F. Toninelli, Gaussian Fluctuations for the stochastic Burgers equation in dimension d≥ 2, CMP, 2024], where the same equation was studied in the weak-coupling regime. Furthermore this model can be seen as a continuous analogue to [H.T. Yau, (log t)²/3 law of the two dimensional asymmetric simple exclusion process, Annals of Mathematics, 2004].