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

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.