相关论文: Metric viscosity solutions and distance-like funct…

This paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the…

Dynamic programming equations for mean field control problems with a separable structure are Eikonal equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity…

On a complete, connected, locally compact, non-compact geodesic space $(X,d)$, we assign each compact set a distance-like function. With the help of these functions, we obtain a pseudo-metric on the space of (non-empty) compact subsets of…

In this note, we provide a smooth variational principle on Wasserstein space by constructing a smooth gauge-type function using the sliced Wasserstein distance. This function is a crucial tool for optimization problems and in viscosity…

This paper is a companion paper to [Lipman and Daubechies 2011]. We provide numerical procedures and algorithms for computing the alignment of and distance between two disk type surfaces. We provide a convergence analysis of the discrete…

This paper is a continuation of our accompanying paper [Talbi, Touzi and Zhang (2021)], where we characterized the mean field optimal stopping problem by an obstacle equation on the Wasserstein space of probability measures, provided that…

In this paper we prove the equivalence between some known notions of solutions to the eikonal equation and more general analogs of the Hamilton-Jacobi equations in complete and rectifiably connected metric spaces. The notions considered are…

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations $F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold $M$.…

The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton-Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the…

In this paper, we study isometries of $p$-Wasserstein spaces. In our first result, for every complete and separable metric space $X$ and for every $p\geq1$, we construct a metric space $Y$ such that $X$ embeds isometrically into $Y$, and…

We show that Busemann functions on a smooth, non-compact, complete, boundaryless, connected Riemannian manifold are viscosity solutions with respect to the Hamilton-Jacobi equation determined by the Riemannian metric and consequently they…

We study the geometry of the space of measures of a compact ultrametric space X, endowed with the L^p Wasserstein distance from optimal transportation. We show that the power p of this distance makes this Wasserstein space affinely…

We establish a comparison principle for viscosity solutions of a class of nonlinear partial differential equations posed on the space of nonnegative finite measures, thereby extending recent results for PDEs defined on the Wasserstein space…

In this paper we introduce a Wasserstein-type distance on the set of Gaussian mixture models. This distance is defined by restricting the set of possible coupling measures in the optimal transport problem to Gaussian mixture models. We…

In this work we prove an analogue, for partial differential equations on the space of probability measures, of the classical vanishing viscosity result known for equations on the Euclidean space. Our result allows in particular to show that…

We prove a comparison result for viscosity solutions of second order parabolic partial differential equations in the Wasserstein space. The comparison is valid for semisolutions that are Lipschitz continuous in the measure in a…

Let $(X,d,\mathfrak{m})$ be a metric measure space. The study of the Wasserstein space $(\mathbb{P}_p(X),\mathbb{W}_p)$ associated to $X$ has proved useful in describing several geometrical properties of $X.$ In this paper we focus on the…

We study the structure of the support of a doubling measure by analyzing its self-similarity properties, which we estimate using a variant of the $L^1$ Wasserstein distance. We show that measure satisfying certain self-similarity conditions…

In this paper we introduce a notion of viscosity solutions for Eikonal equations defined on topological networks. Existence of a solution for the Dirichlet problem is obtained via representation formulas involving a distance function…

We study analogs of value functions arising in classical mechanics in the space of probability measures endowed with the Wasserstein metric $W_p$, for $1<p<\infty$. Our main result is that each of these generalized value functions is a type…