Equations on Wasserstein Space and Applications to Stochastic Mean Field Control and Games

The study of controlled multi-agent systems has seen increased interest in recent years, due to their ubiquity in applications coming from macroeconomics, social behavior, and telecommunications. When the number of agents becomes large, the model can be formally replaced by one involving a mean-field description of the population, analogously to similar models in statistical physics. Justifying this continuum limit, which can be interpreted as a generalized law of large numbers, is often nontrivial and is sensitive to the type of stochastic noise influencing the population, i.e. idiosyncratic or systemic. I will describe a number of settings for which the convergence to mean field stochastic control problems can be resolved through the analysis of a certain Hamilton-Jacobi-Bellman equation posed on the space of probability measures. In particular, we develop new stability and regularity results for the equation, relying on the measurement of the maximum mean discrepancy between probability laws with respect to a suitable reproducing kernel Hilbert space. The results obtained for the equation are general enough to allow for new convergence results for mean field zero-sum differential games.

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