  We consider the Burgers equation posed on the outer communication region of a
Schwarzschild black hole spacetime. Assuming spherical symmetry for the fluid
flow under consideration, we study the propagation and interaction of shock
waves under the effect of random forcing. First of all, considering the initial
and boundary value problem with boundary data prescribed in the vicinity of the
horizon, we establish a generalization of the Hopf--Lax--Oleinik formula, which
takes the curved geometry into account and allows us to establish the existence
of bounded variation solutions. To this end, we analyze the global behavior of
the characteristic curves in the Schwarzschild geometry, including their
behavior near the black hole horizon. In a second part, we investigate the
long-term statistical properties of solutions when a random forcing is imposed
near the black hole horizon and study the ergodicity of the fluid flow under
consideration. We prove the existence of a random global attractor and, for the
Burgers equation outside of a Schwarzschild black hole, we are able to validate
the so-called `one-force-one-solution' principle. Furthermore, all of our
results are also established for a pressureless Euler model which consists of
two balance laws and includes a transport equation satisfied by the integrated
fluid density.
