It is well known in the mean field game literature that certain monotonicity condition is crucial for the uniqueness of mean field equilibria and for the wellposedness of the associated master equation. One interesting observation is that the propagation of the monotonicity (either in Lasry-Lions sense or in displacement sense) of the value function plays the key role here. We shall introduce a method to find conditions on the coefficients which ensure that any solution of the master equation will maintain the monotonicity property. This method also allows us to consider anti-monotonicity and obtain the desired wellposedness provided the coefficients are sufficient anti-monotone in appropriate sense. We finally extend our results to mean field game of controls. This talk is based on a joint work with Gangbo-Meszaros-Zhang and two joint works with Zhang.
Chenchen Mou is currently an assistant professor at City University of Hong Kong. He finished his Ph.D. at Georgia Tech at 2016 and did his Postdoc at UCLA during 2016-2020. He has papers published or to be published in Memoirs of the AMS, Ann. Probab., Ann. Appl. Probab., Comm. Math. Phys., Trans. Amer. Math. Soc. and etc.