  We obtain solutions to conservation laws under any random initial conditions
that are described by Gaussian stochastic processes (in some cases
discretized). We analyze the generalization of Burgers' equation for a smooth
flux function $H\left( p\right) =\left\vert p\right\vert ^{j}$ for $j\geq2$
under random initial data. We then consider a piecewise linear, non-smooth and
non-convex flux function paired with general discretized Gaussian stochastic
process initial data. By partitioning the real line into a finite number of
points, we obtain an exact expression for the solution of this problem. From
this we can also find exact and approximate formulae for the density of shocks
in the solution profile at a given time $t$ and spatial coordinate $x$. We
discuss the simplification of these results in specific cases, including
Brownian motion and Brownian bridge, for which the inverse covariance matrix
and corresponding eigenvalue spectrum have some special properties. We
calculate the transition probabilities between various cases and examine the
variance of the solution $w\left(x,t\right)$ in both $x$ and $t$. We also
describe how results may be obtained for a non-discretized version of a
Gaussian stochastic process by taking the continuum limit as the partition
becomes more fine.
