Homogenization for a Class of Integral Functionals in Spaces of Probability Measures
نویسنده
چکیده
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(R). We introduce an appropriate topology on this set and obtain the existence of a Γ-limit of the rescaled Lagrangians. Our main goal is provide to a representation formula for these Γ-limits in terms of the effective Lagrangians. Not only does this allow us to study “convexity properties” of the effective Lagrangian, but also the differentiability properties of its Legendre transform restricted to constant functions. In 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 wel l as the asymptotic behavior of calibrated curves. Whereas our study in the one-dimensional case covers a large class of Lagrangians, that of 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.
منابع مشابه
A Representation for Characteristic Functionals of Stable Random Measures with Values in Sazonov Spaces
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