  We consider two-player zero-sum differential games (ZSDGs), where the state
process (dynamical system) depends on the random initial condition and the
state process's distribution, and the objective functional includes the state
process's distribution and the random target variable. Unlike ZSDGs studied in
the existing literature, the ZSDG of this paper introduces a new technical
challenge, since the corresponding (lower and upper) value functions are
defined on $\mathcal{P}_2$ (the set of probability measures with finite second
moments) or $\mathcal{L}_2$ (the set of random variables with finite second
moments), both of which are infinite-dimensional spaces. We show that the
(lower and upper) value functions on $\mathcal{P}_2$ and $\mathcal{L}_2$ are
equivalent (law invariant) and continuous, satisfying dynamic programming
principles. We use the notion of derivative of a function of probability
measures in $\mathcal{P}_2$ and its lifted version in $\mathcal{L}_2$ to show
that the (lower and upper) value functions are unique viscosity solutions to
the associated (lower and upper) Hamilton-Jacobi-Isaacs equations that are
(infinite-dimensional) first-order PDEs on $\mathcal{P}_2$ and $\mathcal{L}_2$,
where the uniqueness is obtained via the comparison principle. Under the Isaacs
condition, we show that the ZSDG has a value.
