Well-posedness and large deviations for the obstacle problem of first-order stochastic conservation laws

Abstract

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 $L^1$-contraction and uniqueness under stronger spatial regularity. Moreover, we prove a Freidlin--Wentzell large deviation principle in $L^1(0,T;L^1(\mathbb{T}^N))$. 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 $H^1$-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).