Minimax Theory and its Applications 05 (2020), No. 2, 305--326
Copyright Heldermann Verlag 2020
Hypoelliptic Mean Field Games -- a Case Study
Ermal Feleqi
Dept. of Mathematics, University of Vlora, Vlore 9401, Albania
ermal.feleqi@univlora.edu.al
Diogo Gomes
King Abdullah University of Science and Technology, CEMSE Division, Thuwal 23955-6900, Saudi Arabia
diogo.gomes@kaust.edu.sa
Teruo Tada
King Abdullah University of Science and Technology, CEMSE Division, Thuwal 23955-6900, Saudi Arabia
teruo.tada@kaust.edu.sa
We study hypoelliptic mean-field games (MFG) that arise in stochastic control problems of degenerate diffusions. Here, we consider MFGs with quadratic Hamiltonians and prove the existence and uniqueness of solutions. Our main tool is the Hopf-Cole transform that converts the MFG into an eigenvalue problem. We prove the existence of a principal eigenvalue and a positive eigenfunction, which are then used to construct the unique solution to the original MFG.
Keywords: Mean-Field Games, stationary problems, hypoelliptic operator, eigenvalue problems.
MSC: 35H10, 49L25, 91A13.
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