Well-posedness of the non-local conservation law by stochastic perturbation

Stochastic non-local conservation law equation in the presence of discontinuous flux functions is considered in an L1∩L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{1}\cap L^{2}$$\end{document} setting. The flux function is assumed bounded and integrable (spatial variable). Our result is to prove existence and uniqueness of weak solutions. The solution is strong solution in the probabilistic sense. The proofs are constructive and based on the method of characteristics (in the presence of noise), Itô–Wentzell–Kunita formula and commutators. Our results are new , to the best of our knowledge, and are the first nonlinear extension of the seminar paper (Flandoli et al. in Invent Math 180:1–53, 2010) where the linear case was addressed.