TY  - JOUR
T1  - Bellman systems with mean field dependent dynamics
A1  - Bensoussan, Alain
A1  - Bulíček, Miroslav
A1  - Frehse, Jens
JA  - Chin. Ann. Math. Ser. B
Y1  - 2018
VL  - 39
IS  - 3
SP  - 461
EP  - 486
M2  - doi: 10.1007/s11401-018-0078-4
KW  - Bellman equation
KW  - maximum principle
KW  - mean filed equation
KW  - nonlinear elliptic equations
KW  - Stochastic games
KW  - weak solution
N2  - We deal with nonlinear elliptic and parabolic systems that are the Bellman like systems associated to stochastic differential games with mean field dependent dynamics. The key novelty of the paper is that we allow heavily mean field dependent dynamics. This in particular leads to a system of PDE's with critical growth, for which it is rare to have an existence and/or regularity result. In the paper, we introduce a structural assumptions that cover many cases in stochastic differential games with mean filed dependent dynamics for which we are able to establish the existence of a weak solution. In addition, we present here a completely new method for obtaining the maximum/minimum principles for systems with critical growths, which is a starting point for further existence and also qualitative analysis.
M1  - Preprint project = NCMM
M1  - Preprint year = 2017
M1  - Preprint number = 07
M1  - Preprint ID = NCMM/2017/07
ER  -