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Studying continuous time counterpart of some discrete time dynamics is now astandard and fruitful technique, as some properties hold in both setups. Ingame theory, this is usually done by considering differential games onEuclidean spaces. This allows to infer properties on the convergence of valuesof a repeated game, to deal with the various concepts of approachability, etc.In this paper, we introduce a specific but quite abstract differential gamedefined on the Wasserstein space of probability distributions and we prove theexistence of its value. Going back to the discrete time dynamics, we deriveresults on weak approachability with partial monitoring: we prove that any setsatisfying a suitable compatibility condition is either weakly approachable orweakly excludable. We also obtain that the value for differential games withnonanticipative strategies is the same that those defined with a new concept ofstrategies very suitable to make links with repeated games.

A differential game on Wasserstein space. Application to weak approachability with partial monitoring