  We study the inviscid Burgers equation on the circle
$\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ forced by the derivative of a Poisson point
process on $\mathbb{R}\times\mathbb{T}$. We construct global solutions with
mean $\theta$ simultaneously for all $\theta\in\mathbb{R}$, and in addition
construct their associated global shocks (which are unique except on a
countable set of $\theta$). We then show that as $\theta$ changes, the solution
only changes through the movement of the global shock, and give precise
formulas for this movement. This is an analogue of previous results by the
author and Yu Gu in the viscous case with white-in-time forcing, which related
the derivative of the solution in $\theta$ to the density of a particle
diffusing in the Burgers flow.
