Markovian solutions of inviscid Burgers equation

For solutions of (inviscid, forceless, one dimensional) Burgers equation with random initial condition, it is heuristically shown that a stationary Feller-Markov property (with respect to the space variable) at some time is conserved at later times, and an evolution equation is derived for the infinitesimal generator. Previously known explicit solutions such as Frachebourg-Martin's (white noise initial velocity) and Carraro-Duchon's L\'evy process intrinsic-statistical solutions (including Brownian initial velocity) are recovered as special cases.