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In this paper we study second order master equations arising from mean fieldgames with common noise over arbitrary time duration. A classical solutiontypically requires the monotonicity condition (or small time duration) andsufficiently smooth data. While keeping the monotonicity condition, our goal isto relax the regularity of the data, which is an open problem in theliterature. In particular, we do not require any differentiability in terms ofthe measures, which prevents us from obtaining classical solutions. We shallpropose three weaker notions of solutions, named as {\it good solutions}, {\itweak solutions}, and {\it weak-viscosity solutions}, respectively, andestablish the wellposedness of the master equation under all three notions. Weemphasize that, due to the game nature, one cannot expect comparison principleeven for classical solutions. The key for the global (in time) wellposedness isthe uniform a priori estimate for the Lipschitz continuity of the solution inthe measures. The monotonicity condition is crucial for this uniform estimateand thus is crucial for the existence of the global solution, but is not neededfor the uniqueness in such Lipschitz class. To facilitate our analysis, weconstruct a smooth mollifier for functions on Wasserstein space, which is newin the literature and is interesting in its own right.  Following the same approach of our wellposedness results, we prove theconvergence of the Nash system, a high dimensional system of PDEs arising fromthe corresponding $N$-player game, under mild regularity requirements. We shallalso prove a propagation of chaos property for the associated optimaltrajectories.

Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data