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Journal Articles Pure and Applied Analysis Year : 2023

A geometric Laplace method

Abstract

A classical tool for approximating integrals is the Laplace method. The first-order, as well as the higher-order Laplace formula is most often written in coordinates without any geometrical interpretation. In this article, motivated by a situation arising, among others, in optimal transport, we give a geometric formulation of the first-order term of the Laplace method. The central tool is the Kim-McCann Riemannian metric which was introduced in the field of optimal transportation. Our main result expresses the first-order term with standard geometric objects such as volume forms, Laplacians, covariant derivatives and scalar curvatures of two different metrics arising naturally in the Kim-McCann framework. Passing by, we give an explicitly quantified version of the Laplace formula, as well as examples of applications.

Dates and versions

hal-03911149 , version 1 (22-12-2022)

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Flavien Léger, François-Xavier Vialard. A geometric Laplace method. Pure and Applied Analysis, 2023, 5 (4), pp.1041-1080. ⟨10.2140/paa.2023.5.1041⟩. ⟨hal-03911149⟩
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