  We find a representation of smooth solutions to the Cauchy problem for a
scalar multidimensional conservation law as small diffusion limit of a
stochastic perturbation along characteristics. It helps, in particular, to
study the process of singularities 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 looses its
smoothness. Before this instant the system coincides partly with the one known
as pressure free gas dynamics.
