  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.
