The 2d stochastic Burgers equation is logarithmically superdiffusive with sharp constant (3/2π)^{2/3} λ^{4/3}, and under superdiffusive rescaling it converges to an explicit anisotropic linear stochastic heat equation.
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Superdiffusive Central Limit Theorem for the Stochastic Burgers Equation at the critical dimension
The 2d stochastic Burgers equation is logarithmically superdiffusive with sharp constant (3/2π)^{2/3} λ^{4/3}, and under superdiffusive rescaling it converges to an explicit anisotropic linear stochastic heat equation.