发表机构
University of Zagreb; University of Oslo; University of Montenegro（萨格勒布大学; 奥斯陆大学; 黑山大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了随机标量守恒律有界动力学解存在唯一强初迹，无需初值或非退化条件，通过路径wise紧性论证和随机-Fubini构造处理退化通量，并确保迹的可测性与收敛性。
AI 中文摘要
我们证明了随机标量守恒律的每个有界动力学解都存在唯一的强初迹，尽管没有给定初值，也没有对通量施加非退化条件。证明的核心是路径wise的。在固定一个实现后，重标度的随机项和动力学测度在爆破极限中消失，因此Panov的紧性论证适用；退化通量区间通过递归降维处理。随机环境带来了几个额外的困难。鞅恒等式必须在所有约化过程中在一个共同的满概率集合上保持有效，这需要参数化的随机-Fubini构造。此外，路径wise迹不自动可测，因为其例外集可能依赖于实现。确定性时间平均和滤过的右连续性产生了一个联合可测的初迹，在本质时间上具有局部强收敛性，既几乎必然又依均值收敛。
英文摘要
We prove existence and uniqueness of a strong initial trace for every bounded kinetic solution of a stochastic scalar conservation law, although no initial value is prescribed and no nondegeneracy condition is imposed on the flux. The core of the proof is pathwise. After fixing a realization, the rescaled stochastic terms and kinetic measure vanish in the blow-up limit, so Panov's compactness argument applies; degenerate flux intervals are handled by recursive dimension reduction. The stochastic setting creates several additional difficulties. The martingale identities must remain valid on one common full-probability set throughout the reductions, which requires a parameterized stochastic-Fubini construction. Moreover, the pathwise trace is not automatically measurable because its exceptional sets may depend on the realization. Deterministic time averages and right-continuity of the filtration yield a jointly measurable initial trace, with local strong convergence along essential times, both almost surely and in mean.
