Geodesic
Abstract McKean--Vlasov and Hamilton--Jacobi--Bellman Equations, Their Fractional Versions and Related Forward--Backward Systems on Riemannian Manifolds
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 128-150

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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.
Keywords: Fractional McKean–Vlasov type equations on manifolds, fractional Hamilton–Jacobi–Bellman–Isaacs equations on manifolds, fractional forward–backward systems on manifolds, dual Banach triples, mild solutions, Caputo–Dzherbashyan fractional derivative, smoothing and smoothness preserving operator semigroups. 
@article{TM_2021_315_a8,
     author = {V. N. Kolokoltsov and M. S. Troeva},
     title = {Abstract {McKean--Vlasov} and {Hamilton--Jacobi--Bellman} {Equations,} {Their} {Fractional} {Versions} and {Related} {Forward--Backward} {Systems} on {Riemannian} {Manifolds}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {128--150},
     publisher = {mathdoc},
     volume = {315},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2021_315_a8/}
}
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V. N. Kolokoltsov; M. S. Troeva. Abstract McKean--Vlasov and Hamilton--Jacobi--Bellman Equations, Their Fractional Versions and Related Forward--Backward Systems on Riemannian Manifolds. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 128-150. http://geodesic.mathdoc.fr/item/TM_2021_315_a8/