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 $x=L$.