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