AI 中文总结
研究高速公路交通波自发形成的随机模型，基于含不连续、依赖梯度通量的守恒律构建，证明了随机解相关量的界，还表明以守恒律随机解为路径的马尔可夫过程有唯一平稳概率分布且遍历。
AI 中文摘要
本文介绍了一种高速公路上交通波自发形成的随机模型。它依据具有不连续、依赖梯度通量的守恒律来构建。在不稳定状态下，解的非唯一性使得交通密度在随机点和随机时间出现N形尖峰。证明了随机解总变差的期望值和冲击次数的界，还得到了沿给定高速公路路段汽车平均速度和平均加速度的界。最后证明了以守恒律随机解为路径的马尔可夫过程存在唯一平稳概率分布且遍历。
英文摘要
The paper introduces a stochastic model for the spontaneous formation of traffic waves on a highway. This is formulated in terms of a conservation law with discontinuous, gradient-dependent flux. In an unstable regime, the non-uniqueness of solutions allows for the emergence of $N$-shaped spikes in the traffic density, at random points and times. Bounds are proved on the expected value of the total variation of the random solution and on the expected number of shocks. Further bounds are obtained on the average velocity and on the expected average acceleration of cars, along a given stretch of highway. Finally, it is proved that the Markov process, whose paths are random solutions to the conservation law, admits a unique stationary probability distribution and is ergodic.
