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We consider the Burgers equation posed on the outer communication region of aSchwarzschild black hole spacetime. Assuming spherical symmetry for the fluidflow under consideration, we study the propagation and interaction of shockwaves under the effect of random forcing. First of all, considering the initialand boundary value problem with boundary data prescribed in the vicinity of thehorizon, we establish a generalization of the Hopf--Lax--Oleinik formula, whichtakes the curved geometry into account and allows us to establish the existenceof bounded variation solutions. To this end, we analyze the global behavior ofthe characteristic curves in the Schwarzschild geometry, including theirbehavior near the black hole horizon. In a second part, we investigate thelong-term statistical properties of solutions when a random forcing is imposednear the black hole horizon and study the ergodicity of the fluid flow underconsideration. We prove the existence of a random global attractor and, for theBurgers equation outside of a Schwarzschild black hole, we are able to validatethe so-called `one-force-one-solution' principle. Furthermore, all of ourresults are also established for a pressureless Euler model which consists oftwo balance laws and includes a transport equation satisfied by the integratedfluid density.

Ergodicity of spherically symmetric fluid flows outside of a Schwarzschild black hole with random boundary forcing