The law of the iterated logarithm for the solution of the Burgers equation with random initial data (Q1966268)

It is proved that for the solution \(v(t,x)\) of \(\frac{\partial v}{\partial t} + (v,\nabla)v=\kappa\nabla v\), \(v(0,x)=-2\kappa\nabla\xi(x)\), \((t,x)\in \mathbb{R}^1_+\times \mathbb{R}^1\) with \(\xi\) a zero-range shot noise the relation \(\varlimsup_{t\rightarrow\infty} \frac{|v(t,x)|}{ct^{3/4}(\ln\ln t)^{1/2}} =1\) (almost everywhere) holds.