Scalar conservation laws with white noise initial data

The statistical description of the scalar conservation law form $\rho_t=H(\rho)_x$ with $H: \mathbb{R} \rightarrow \mathbb{R}$ a smooth convex function has been an object interest when initial profile $\rho(\cdot,0)$ is random. special case $H(\rho)=\frac{\rho^2}{2}$ (Burgers equation) in particular received extensive past and now understood for various random conditions. We solve this paper conjecture on solution at any time $t>0$ general class hamiltonians $H$ show that it stationary piecewise-smooth Feller process. Along way, we study excursion process two-sided linear Brownian motion $W$ below strictly $\phi$ superlinear growth derive generalized Chernoff distribution variable $\text{argmax}_{z \in \mathbb{R}} (W(z)-\phi(z))$. Finally, white noise derived from abrupt L\'evy process, shocks structure a.s discrete fixed under some mild assumptions $H$.