ИСТИНА

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. Related questions concern the study of Hilbert-space valued Mc-Kean-Vlasov diffusion. The estimates of growth are shown to be expressed (even for non-fractional diffusions) in terms of the Mittag-Leffler and Le Roy functions. (Based on the joint work with M.S. Troeva).