Moment-sos hierarchy and exit location of stochastic processes

The moment sum of squares (moment -SOS) hierarchy produces sequences of upper and lower bounds on functionals of the exit time location of a polynomial stochastic differential equation with polynomial constraints, at the price of solving semidefinite optimization problems of increasing size. In this note we use standard results from elliptic partial differential equation analysis to prove convergence of the bounds produced by the hierarchy. We also use elementary convex analysis to describe a super- and sub -solution interpretation dual to a linear formulation on occupation measures. Finally, we introduce a novel Christoffel-Darboux approach for the recovery of the exit location and occupation measures. The practical relevance of these results is illustrated with numerical examples.