Workshop on deterministic and stochastic partial differential equations

The Master equation is an infinite dimensional partial differential equation in a state space comprising Euclidean vectors and probability measures. It was introduced by Lasry and Lions for the study of Mean Field Games. We derive this Master Equation from a special Ito formula based on a non-standard differential calculus for functions of probability measures. If time permits, we shall also derive the Master Equation for Mean Field Stochastic Control problems (i.e. the control of McKean-Vlasov stochastic differential equations). Nicolai V. Krylov (University of Minnesota) Title: On the existence of W 2 p solutions for fully nonlinear elliptic equations under relaxed convexity assumptions Abstract: We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like H(v,Dv,Dv, x) = 0 in smooth domains without requiring H to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of H at points at which |Dv| ≤ K, where K is any given constant. For large |Dv| some kind of relaxed convexity assumption with respect to Dv mixed with a VMO condition with respect to x are still imposed. The solutions are sought in Sobolev classes. We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like H(v,Dv,Dv, x) = 0 in smooth domains without requiring H to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of H at points at which |Dv| ≤ K, where K is any given constant. For large |Dv| some kind of relaxed convexity assumption with respect to Dv mixed with a VMO condition with respect to x are still imposed. The solutions are sought in Sobolev classes.