A Bhatnagar-Gross-Krook Approximation to Stochastic Scalar Conservation Laws

and study its approximation in the sense of Bhatnagar-Gross-Krook (a BGKlike approximation for short). In particular, we aim to describe the conservation law (1.1) as the hydrodynamic limit of the stochastic BGK model, as the microscopic scale ε goes to 0. The literature devoted to the deterministic counterpart, i.e. corresponding to the situation Φ = 0, is quite extensive (see [1], [11], [15], [16], [17], [18], [19], [20]). In that case, the BGK model is given as follows