| Abstract: |
| 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, which 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. |
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