Burgers' equation has frequently been considered with initial data which is a stochastic process. Sinai (1992) investigated white noise initial data. Carraro and Duchon (1994) noted that a Levy process at time zero should lead to a Levy process at fixed later time, an observation made rigorous for Cole-Hopf solutions by Bertoin (1998). In 2010, Menon and Srinivasan conjectured that more general scalar conservation laws preserve certain Feller processes, and described how the generator for the spatial processes should evolve in time. We discuss the speaker's dissertation work (supervised by F. Rezakhanlou) verifying an analogue of Menon and Srinivasan's conjecture for bounded, piecewise constant initial data.