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 prove 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 Lévy process, we show that the structure of shocks of the solution is a.s discrete at any fixed time \(t>0\) under some mild assumptions on H.

\(\rho _t=H(\rho )_x\)形式的标量守恒定律的统计描述，其中\(H: {\mathbb {R}} \rightarrow {\mathbb {R}}\)平滑当初始轮廓\(\rho (\cdot ,0)\)是随机的时，凸函数一直是一个感兴趣的对象。当特殊情况\（H（\ RHO）= \压裂{\ RHO ^ 2} {2} \）（Burgers方程）已经在特定的接收的广泛关注在过去和现在被理解为各种随机的初始条件。我们在本文中证明了对一般类哈密顿量H在任何时间\(t>0\)的解的轮廓的猜想并证明它是一个平稳的分段平滑 Feller 过程。在此过程中，我们研究了具有超线性增长的任何严格凸函数\(\phi \)下的两侧线性布朗运动W的偏移过程，并推导出随机变量\(\text {argmax}_ {z \in {\mathbb {R}}} (W(z)-\phi (z))\)。最后，当\(\rho (\cdot ,0)\)是源自突然Lévy 过程的白噪声时，我们表明解的激波结构在任何固定时间都是离散的\(t>0\)在对H 的一些温和假设下。