Some statistical properties of the Burgers equation with white-noise
  initial velocity
  We revisit the one-dimensional Burgers equation in the inviscid limit for
white-noise initial velocity. We derive the probability distributions of
velocity and Lagrangian increments, measured on intervals of any length $x$.
This also gives the velocity structure functions. Next, for the case where the
initial density is uniform, we obtain the distribution of the density, over any
scale $x$, and we derive the density two-point correlation and power spectrum.
Finally, we consider the Lagrangian displacement field and we derive the
distribution of increments of the Lagrangian map. We check that this gives back
the well-known mass function of shocks. For all distributions we describe the
limiting scaling functions that are obtained in the large-scale and small-scale
limits. We also discuss how these results generalize to other initial
conditions, or to higher dimensions, and make the connection with a heuristic
multifractal formalism. We note that the formation of point-like masses
generically leads to a universal small-scale scaling for the density
distribution, which is known as the ``stable-clustering ansatz'' in the
cosmological context (where the Burgers dynamics is also known as the
``adhesion model'').
