Abstract McKean-Vlasov and Hamilton-Jacobi-Bellman equations, their fractional versions and related forward-backward systems on a Riemannian manifold

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-Isaaks equation of stochastic control and games thus allowing for a unified analysis of these equations. Looking at these equations as evolving in dual Banach triples allows us to recast directly the properties of one type to the properties of another type leading to an effective theory of coupled forward-backward systems (forward McKean-Vlasov evolution and backward HJB evolution) that are central to the modern theory of mean-field games. We are working with the mild solutions to the fractional nonlinear equations that are based on the Zolotarev integral representation for the Mittag-Leffler functions. The abstract setting developed allows us to include in our analysis the related nonlinear fractional equations and forward-backward systems on manifolds yielding results that are possibly new even for classical (not fractional) equations. We obtain the basic well-poesdness results for these equations, and prove their continuous and smooth dependence on the initial data.