2026/07/30 by Ruoyang Liu, Rangrang Zhang
Mathematics · #math.PR #math.AP #msc:60H15 #msc:35R60 #msc:60F10 #msc:35R35
61 pages
arxiv created 2026/07/30 · arxiv updated 2026/07/31
This paper studies the obstacle problem for first-order scalar conservation laws driven by multiplicative noise. By adapting a barrier-substitution strategy to the kinetic formulation, we permit the reflection measure to be a general Radon measure and eliminate the usual obstacle-noise compatibility condition. We establish the existence of kinetic solutions for continuous obstacles and further derive L1-contraction and uniqueness under stronger spatial regularity. Moreover, we prove a Freidlin--Wentzell large deviation principle in L1(0,T;L1(\mathbbTN)). Unlike existing large-deviation arguments based on control-uniform penalization of the obstacle, we work directly with the reflected skeleton equation and retain the possibly singular Radon reflection measure throughout, while a viscous approximation provides the spatial H1-regularity required for compactness before passing to the inviscid limit. Our results provide a hyperbolic counterpart to the large deviation theory for parabolic reflected SPDEs developed by Matoussi, Sabbagh, and Zhang (2021).