Stochastic Analysis for the Complex Monge-Ampère Equation. (An Introduction to Krylov's Approach.)

We here gather in a single note several original probabilistic works devoted to the analysis of the C^(1,1) regularity of the solution to the possibly degenerate complex Monge-Ampère equation. The whole analysis relies on a probabilistic writing of the solution as the value function of a stochastic optimal control problem. Such a representation has been introduced by Gaveau in the late 70's and used in an exhaustive way by Krylov in a series of papers published in the late 80's. All the arguments we here use follow from these seminal works.