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### Completion of spaces of measures w.r.t. weak norms

For the sake of concreteness denote by $M_0(X)$ the linear space of all signed Borel measures $\sigma$ with $\sigma(X)=0$ on some metric space $(X,d)$ and fix some base point $x_0\in X$. On this space ...
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### Existence of optimal coupling in optimal transport

Let $P,Q$ be any two distributions over a space $\mathcal{X}$ and let $\mathcal{M}(P,Q)$ be the set of all couplings of $P$ and $Q.$ For a given metric $d$ over $\mathcal{X},$ the optimal transport ...
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### Is there an elementary proof of the polar factorization theorem for vector-valued function?

I have recently learned the polar factorization theorem for vector-valued functions due to Brenier. Namely, given a probability space $(X,\mu)$ and a bounded domain $\Omega\subset \mathbb{R}^n$ with ...
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### $\mathbb{P}(d(X,Y)>\alpha)<\beta$ if $\mathbb{P}(X\in E)\leq \mathbb{P}(Y\in E^{\alpha})+\beta$ for all measurable E

Given two random variables X,Y with measures P,Q. Show that if $P(E) \le Q(E^\alpha) + \beta$ for all measurable $E\subset\mathbb{R}$ then $\mathbb{P}(d(X,Y)>\alpha)<\beta$. Only hints please. ...
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### c-superdifferential is unique +cost function is differentiable, then the potential function is differentiable?

Let $M$ be a compact Riemannian manifold, $\mu$ and $\nu$ are two Borel probability measures, the cost function $c(x,y)=\frac{d^2(x,y)}{2}$. It's well known that the infimum of the Kontorovich's ...
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For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$\... 1answer 159 views ### Reference request: Wasserstein metric spaces for non linear weights/mobility? There is a very nice theory of gradient flows in metric spaces by Ambrosio, Gigli and Savaré. One particularly important application is the quadratic Wasserstein setting, where the metric space in ... 0answers 138 views ### Reference request: density of C_c^{\infty}(\mathbb R^d) in L^2(\mathbb R^d,d\rho) My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ... 1answer 230 views ### PDE-Based Triangle Inequality for Optimal Transportation Suppose \Omega is a suitably regular domain in \mathbb{R}^n and \rho_0,\rho_1\in\textrm{Prob}(\Omega). Benamou and Brenier showed that the L_2 transportation distance between \rho_0 and \... 1answer 381 views ### Modulus of of continuity of a convolution operator with respect to Wasserstein metric For a (discrete) measure G on some reasonable metric space \Theta, consider the map G \mapsto f_G defined as$$ f_G := f*G(dx) := \int f(dx|\theta) G(d\theta) $$for some nice kernel function ... 0answers 36 views ### Constructing a family of domains for application of method of continuity in optimal transportation Anyway can help me about this paper? http://arxiv.org/pdf/math/0601086v4.pdf I want to ask page 20, The author want to construct a family of subdomain for using method of continuity. But I can't ... 0answers 600 views ### 1-Wasserstein distance between two multivariate normal The p-Wasserstein between two measures \nu_1 and \nu_2 on X is given by$$d_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$... 1answer 177 views ###$X$Polish geodesic implies$(P_2(X), W_2)$geodesic If$X,d$is a complete and separable space then the space of Borel probability measures with finite second moment on$X$endowed with the Wasserstein distance$W_2$is geodesic. I am looking for a ... 0answers 106 views ### a generalization of Monge-Kantorovich Problem I am thinking about the martingale version of Monge-Kantorovich Problem. Let$\mu(x)$and$\nu(y)$denote two density laws on$\mathbb{R}$, and define$M(\mu,\nu)$the set of densities$f(x,y)$on$\...
I'm interested in (and a bit confused by) the following theorem of Caffarelli, proven in section $4$ of his paper Boundary regularity of maps with convex potentials II: Assume $u$ is a convex ...
One great inside of Felix Otto is that the Wasserstein metric from optimal transportation gives the space of (finite second moment, probability) measures on $\mathbb{R}^n$ (or a manifold) a kind of ...