  We develop a Lagrangian approach to conservation-law anomalies in weak
solutions of inviscid Burgers equation, motivated by previous work on the
Kraichnan model of turbulent scalar advection. We show that the entropy
solutions of Burgers possess Markov stochastic processes of (generalized)
Lagrangian trajectories backward in time for which the Burgers velocity is a
backward martingale. This property is shown to guarantee dissipativity of
conservation-law anomalies for general convex functions of the velocity. The
backward stochastic Burgers flows with these properties are not unique,
however. We construct infinitely many such stochastic flows, both by a
geometric construction and by the zero-noise limit of the Constantin-Iyer
stochastic representation of viscous Burgers solutions. The latter proof yields
the spontaneous stochasticity of Lagrangian trajectories backward in time for
Burgers, at unit Prandtl number. It is conjectured that existence of a backward
stochastic flow with the velocity as martingale is an admissibility condition
which selects the unique entropy solution for Burgers. We also study linear
transport of passive densities and scalars by inviscid Burgers flows. We show
that shock solutions of Burgers exhibit spontaneous stochasticity backward in
time for all Prandtl numbers, implying conservation-law anomalies for linear
transport. We discuss the relation of our results for Burgers with
incompressible Navier-Stokes turbulence, especially Lagrangian admissibility
conditions for Euler solutions and the relation between turbulent cascade
directions and time-asymmetry of Lagrangian stochasticity.
