Specialized effective Positivstellens\"atze for improved convergence rates of the moment-SOS hierarchy

Recently a moment-sum-of-squares hierarchy for exit location estimation of stochastic processes has been presented. When restricting to the special case of the unit ball, we show that the solutions approach the optimal value by a super-polynomial rate. To show this result we state a new effective Positivstellensatz on the sphere with quadratic degree bound based on a recent Positivstellensatz for trigonometric polynomials on the hypercube and pair it with a recent effective Positivstellensatz on the unit ball. At the present example, we aim to highlight the effectiveness of specialized Positivstellens\"atze for the moment-SoS hierarchy and their interplay with problem intrinsic properties.