We will start discussing a new method of how to prove exponential decay for the solutions of the wave equation on a Schwarzschild de Sitter black hole spacetime by exploiting a novel "relatively non-degenerate" estimate. This estimate does not degenerate at trapping. The main ingredient in proving this estimate is to commute with a novel vector field that "sees" trapping. Then, we will discuss a natural generalization of the vector field commutation in Schwarzschild de Sitter to the entire subextremal Kerr de Sitter black hole spacetime, by commuting with a pseudodifferential operator. There are more technicalities because of the elaborate nature of trapping. Time permitting we will discuss how to use this black box estimate to prove stability and exponential decay for the solutions of a quasilinear wave equation on Kerr de Sitter.