  We introduce a class of abstract nonlinear fractional pseudo-differential
equations in Banach spaces that includes both the Mc-Kean-Vlasov-type equations
describing nonlinear Markov processes and the
Hamilton-Jacobi-Bellman(HJB)-Isaacs equation of stochastic control and games
thus allowing for a unified analysis of these equations. This leads to an
effective theory of coupled forward-backward systems (forward McKean-Vlasov
evolution and backward HJB-Isaacs evolution) that are central to the modern
theory of mean-field games.
