  We derive a kinetic equation to describe the statistical structure of
solutions $\rho$ to scalar conservation laws $\rho_t=H(x,t,\rho )_x$, with
certain Markov initial conditions. When the Hamiltonian function is convex and
increasing in $\rho$, we show that the solution $\rho(x,t)$ is a Markov process
in $x$ (respectively $t$) with $t$ (respectively $x$) fixed. Two classes of
Markov conditions are considered in this article. In the first class, the
initial data is characterize by a drift $b$ which satisfies a linear PDE, and a
jump density $f$ which satisfies a kinetic equation as time varies. In the
second class, the initial data is a concatenation of fundamental solutions that
are characterized by a parameter $y$, which is a Markov jump process with a
jump density $g$ satisfying a kinetic equation. When $H$ is not increasing in
$\rho$, the restriction of $\rho$ to a line in $(x,t)$ plane is a Markov
process of the same type, provided that the slope of the line satisfies an
inequality.
