  We study Lagrangian trajectories and scalar transport statistics in decaying
Burgers turbulence. We choose velocity fields, solutions of the inviscid
Burgers equation, whose probability distributions are specified by Kida's
statistics. They are time-correlated, not time-reversal invariant and not
Gaussian. We discuss in some details the effect of shocks on trajectories and
transport equations. We derive the inviscid limit of these equations using a
formalism of operators localized on shocks. We compute the probability
distribution functions of the trajectories although they do not define Markov
processes. As physically expected, these trajectories are statistically
well-defined but collapse with probability one at infinite time. We point out
that the advected scalars enjoy inverse energy cascades. We also make a few
comments on the connection between our computations and persistence problems.
