  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.
