Homogenization for a class of integral functionals in spaces of probability measures

We study the homogenization of a class of actions with an underlying Lagrangian L defined on the set of absolutely continuous paths in the Wasserstein space Pp(Rd). We introduce an appropriate topology on this set and obtain the existence of a Γ-limit of the rescaled Lagrangians. Our main goal is to provide a representation formula for these Γ-limits in terms of the effective Lagrangians. This allows us to study not only the “convexity properties” of the effective Lagrangian, but also the differentiability properties of its Legendre transform restricted to constant functions. For the case d>1 we obtain partial results in terms of an effective Lagrangian defined on Lp((0,1)d;Rd). Our study provides a way of computing the limit of a family of metrics on the Wasserstein space. The results of this paper can also be applied to study the homogenization of variational solutions of the one-dimensional Vlasov-Poisson system, as well as the asymptotic behavior of calibrated curves (Fathi (2003) [6], Gangbo and Tudorascu (2010) [12]). Whereas our study for the one-dimensional case covers a large class of Lagrangians, that for the higher dimensional case is concerned with special Lagrangians such as the ones obtained by regularizing the potential energy of the d-dimensional Vlasov-Poisson system.