Stochastic and Multiscale Modeling and Computation Seminar by Fanze Kong: Ground States in Several Ergodic Stationary Mean-Field Games Systems

Abstract: Motivated by the theory of statistical physics, Huang et al. and Lasry et al. in 2007 independently proposed a class of strongly coupled PDEs named as Mean-field Games systems (MFGs) to describe the game among a huge number of rational players.  Over the last decade, scholars extensively studied MFGs with the increasing cost by following techniques stated in the pioneering works of Lasry and Lions.  Whereas, the properties of MFGs with the decreasing cost are not understood as well as the case of those with the increasing cost.   

In this talk, we focus on several stationary second order MFGs with the local or nonlocal decreasing costs and mass critical exponents, then discuss the asymptotic behaviors of ground states via the variational method.  First of all, we present the formulation of optimal Gagliardo-Nirenberg type inequalities associated with the potential-free MFGs.  Next, we analyze the critical mass phenomena in MFGs with some mild assumptions imposed on potential functions. Finally, under certain types of potential functions, the basic and refined blow-up behaviors of ground states in some singular limits will be discussed.  In particular, the intricate connections between MFGs and nonlinear Schr\"{o}dinger equations, along with some results of maximal regularities in Hamilton-Jacobi equations subject to superlinear gradient terms, are introduced.

Stochastic and Multiscale Modeling and Computation Seminar