A representation of solutions to a scalar conservation law in several dimensions

We find a representation of smooth solutions to the Cauchy problem for a scalar multidimensional conservation law as a small diffusion limit of a stochastic perturbation along characteristics. It helps, in particular, to study the process of singularity formation. Further, we introduce an associated system of balance laws that can be interpreted as describing the motion of a continuum with some specific pressure term. This term arises only after the instant when the solution to the initial Cauchy problem loses its smoothness. Before this instant the system coincides partly with the one known as pressure free gas dynamics.