In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by L\'evy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of a solution is established. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that L\'evy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. Furthermore, we establish a result on vanishing non-local regularization of scalar stochastic conservation laws.

中文翻译：

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由 Lévy 噪声驱动的分数守恒定律的柯西问题

在本文中，我们探讨了与 L'evy 过程驱动的多维分数守恒定律相关的一些主要数学问题。利用适应的熵公式，建立解的存在性和唯一性的结果。此外，在 L\'evy 噪声仅取决于解的假设下，我们使用有界变异 (BV) 估计值对消失粘度近似值，推导出对熵解非线性度的显式连续依赖估计。该结果用于显示随机消失粘度方法的误差估计。此外，我们建立了一个关于标量随机守恒定律消失的非局部正则化的结果。