$(\log t)^\frac{2}{3}$-superdiffusivity for the 2d stochastic Burgers equation

Abstract

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 $$\partial_t\eta =\frac{1}{2}\Delta\eta+ \mathfrak w\cdot\nabla(\eta^2) + \nabla\cdot\xi,$$ where $\xi$ is $d$-dimensional space time white noise and $\mathfrak w$ 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 $(\log t)^\frac23$, in a Tauberian sense, up to $\log\log\log t$ 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 $d\geq 2$, 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, $(\log t)^\frac{2}{3}$ law of the two dimensional asymmetric simple exclusion process, Annals of Mathematics, 2004].