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6
votes
1answer
217 views

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 ...
0
votes
0answers
32 views

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 ...
4
votes
0answers
65 views

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 ...
1
vote
0answers
106 views

$\mathbb{P}(d(X,Y)>\alpha)<\beta$ 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. ...
0
votes
0answers
31 views

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 ...
4
votes
1answer
219 views

continuity of the Boltzmann entropy in the Wasserstein metric

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 $$ ...
0
votes
0answers
46 views

characterization of the equivalence between two probability measures

Let $X=(X_1,...,X_n)$ be a canonical process defined on the Euclidean space $R^n$, i.e. $X(x)=x$ for all $x\in R^n$ and $\mathbb F=\{\mathcal{F}_k\}_{1\le k\le n}$ be its natural filtration, i.e. ...
4
votes
1answer
153 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 ...
0
votes
0answers
133 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 ...
4
votes
1answer
225 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 ...
2
votes
1answer
376 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 ...
1
vote
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 ...
8
votes
0answers
561 views

1-Wasserstein distance between two multivariate normal

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by ...
2
votes
1answer
176 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 ...
2
votes
0answers
100 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 ...
3
votes
0answers
207 views

Tangential boundary regularity for optimal transport maps

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 ...
9
votes
0answers
238 views

Where to use differential calculus on space of measures?

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 ...