Existence of strong initial traces for stochastic conservation laws

Abstract

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.