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.
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BibTeX
@misc{erceg2026existence,
title = {{Existence of strong initial traces for stochastic conservation laws}},
author = {Marko Erceg and Kenneth Karlsen and Nikola Konatar and Darko Mitrovic},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20735},
eprint = {2609.20735},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20735},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Existence of strong initial traces for stochastic conservation laws
AU - Marko Erceg
AU - Kenneth Karlsen
AU - Nikola Konatar
AU - Darko Mitrovic
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.20735
UR - https://arxiv.org/abs/2609.20735
AB - 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.
ER -