Scalar conservation laws with monotone pure-jump Markov initial conditions

In 2010 Menon and Srinivasan published a conjecture for the statistical structure of solutions (rho ) to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe (rho (x,t)) as a stochastic process in x with t fixed. In this article we verify an analogue of the conjecture for initial conditions which are bounded, monotone, and piecewise constant. Our argument uses a particle system representation of (rho (x,t)) over (0 le x le L) for (L > 0), with a suitable random boundary condition at (x = L).