The statistical description of the scalar conservation law of the form $ρ_t=H(ρ)_x$ with $H: \mathbb{R} \rightarrow \mathbb{R}$ a smooth convex function has been an object of interest when the initial profile $ρ(\cdot,0)$ is random. The special case when $H(ρ)=\frac{ρ^2}{2}$ (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 $φ$ with superlinear growth and derive a generalized Chernoff distribution of the random variable $\text{argmax}_{z \in \mathbb{R}} (W(z)-φ(z))$. Finally, when $ρ(\cdot,0)$ is a white noise derived from an abrupt Lévy 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 $H$.