In this paper, we prove the equivalence of kinetic solutions and entropy solutions for the initial-boundary value problem with a non-homogeneous boundary condition for a multi-dimensional scalar first-order conservation law with a multiplicative noise. We somewhat generalized the definitions of kinetic solutions and of entropy solutions given in Kobayasi and Noboriguchi  and Bauzet, Vallet and Wittobolt , respectively.
"The Equivalence Theorem of Kinetic Solutions and Entropy Solutions for Stochastic Scalar Conservation Laws." Tokyo J. Math. 38 (2) 575 - 587, December 2015. https://doi.org/10.3836/tjm/1452806058