  A statistical theory is developed for the stochastic Burgers equation in the
inviscid limit. Master equations for the probability density functions of
velocity, velocity difference and velocity gradient are derived. No closure
assumptions are made. Instead closure is achieved through a dimension reduction
process, namely the unclosed terms are expressed in terms of statistical
quantities for the singular structures of the velocity field, here the shocks.
Master equations for the environment of the shocks are further expressed in
terms of the statistics of singular structures on the shocks, namely the points
of shock generation and collisions. The scaling laws of the structure functions
are derived through the analysis of the master equations. Rigorous bounds on
the decay of the tail probabilities for the velocity gradient are obtained
using realizability constraints. We also establish that the probability density
function of the velocity gradient has a left power tail with exponent -7/2.
