Path-dependent differential games of inf-sup type and Isaacs partial differential equations

We consider two-person zero-sum differential games under path-dependent dynamics and costs fixing the order of the inf and the sup for the payoffs. Using dynamic programming methods, the inf-sup type value functions are defined on sets of past trajectories of states which are infinite-dimensional spaces. Under a notion of co-invariant derivatives for the infinite-dimensional spaces, we obtain infinitesimal generators of the dynamic programming operators related to the value functions. Then, we associate the inf-sup type value functions with path-dependent Isaacs partial differential equations in the sense of viscosity solutions proposed by N. Lukoyanov.