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 describing Markov processes and Hamilton–Jacobi–Bellman–Isaacs equation stochastic control games. This allows for unified analysis these equations, which leads to an effective theory coupled forward–backward systems (forward evolution backward evolution) are central modern mean-field