论文标题
带有白噪声初始数据的标量保护法
Scalar Conservation Laws with white noise initial data
论文作者
论文摘要
形式的标量保护定律的统计描述$ρ_t= h（ρ）_x $ with $ h：\ mathbb {r} \ rightArrow \ rightarrow \ mathbb {r} $一个平滑的凸功能是当初始配置文件$ρ（\ cdot，0）$是随机的。当$ h（ρ）= \ frac {ρ^2} {2} $（burgers方程）尤其在过去收到了广泛的利益时，特殊情况在过去的各种随机初始条件中被理解。我们在本文中解决了有关解决方案的概况的猜想。一路上，我们研究了以下任何严格的凸函数$ ϕ $的双面线性布朗运动$ W $的偏移过程，并随机变长并得出了随机变量$ \ text {argmax} _ {z {z {z {in \ mathbb {r}}（r}}}（w（z） - （z） - （z） - ϕ（z）））$的Chernoff分布。最后，当$ρ（\ cdot，0）$是从突然的莱维过程中得出的白噪声时，我们表明解决方案的冲击结构是A.S离散的任何固定时间$ t> 0 $在$ h $的某些轻度假设下。
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 $H$.
