Abstract McKean–Vlasov and Hamilton–Jacobi–Bellman Equations, Their Fractional Versions and Related Forward–Backward Systems on Riemannian Manifolds

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