I will discuss some results of a recent paper with Gregory Eyink on 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. I will demonstrate this directly by considering the zero-noise limit of the Constantin-Iyer stochastic representation of viscous Burgers solutions. This proof yields the spontaneous stochasticity of Lagrangian trajectories backward in time for Burgers, at unit Prandtl number, and is the first result of this type for a deterministic PDE problem. The "martingale property" then guarantees dissipativity of conservation-law anomalies for general convex functions of the velocity. 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.
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