Questions tagged [optimal-transportation]
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204
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Schrödinger Bridge for other costs
Stochastic control formulations of the Schrödinger bridge problem between $\mu,\nu$ are well known (e.g Chen et al Eq. 4.23)
$$\inf \limits_{p_t, v_t} \int_0^T \int \frac{1}{2}\lvert v_t\rvert^2 p_t ...
- 71
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1
answer
60
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A variant of (discrete) optimal transport problem
Let $\alpha=(\alpha_1,\ldots,\alpha_m)\subset\mathbb R^m_+$ and $\beta=(\beta_1,\ldots,\beta_n)\subset\mathbb R^n_+$ be given and satisfy
$$\sum_{i=1}^m \alpha_i =1 = \sum_{j=1}^n\beta_j.$$
Define $\...
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Representation formula for the continuity equation on a separable Hilbert space
The following is an informal question for which I'd like to (ideally) find a reference. I'm quite a novice in this area but would be happy to find a reference to a theorem along the following lines (...
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2
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0
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64
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Approximating solutions to Monge-Ampere from optimal transport plans
I am interested in finding numerical solutions to a Monge-Ampere type equation for applications in physics. Due to the close connection between Monge-Ampere and optimal transport and the well ...
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2
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Getting Wasserstein closeness from a derivative estimate
In my setting, $\mu$ and $\nu$ are probability measures on $\mathbb{R}^{2}$ with compact support. For any function $f\in{C^{2}_{b}(\mathbb{R}^{2})}$, I have the estimate:
$$
|\mathbb{E}_{\mu}(f)-\...
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3
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1
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102
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Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
Let $p \in [1, \infty)$. Let $\mathcal P_p(\mathbb R^d)$ be the space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moments. Let $D_p$ be the collection of all Borel measurable ...
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1
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Is the Wasserstein distance to the empirical measure minimized by the underlying distribution?
Let $S$ be a metric space and denote the set of probability measures on $S$ by $\mathcal{P}(S)$. Fix $\mu\in \mathcal{P}(S)$ and denote the law of $N\geq 1$ i.i.d samples $X=(X_1,\ldots,X_N)$ from $\...
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Riemannian submanifolds of $2$-Wasserstein space
In the article "Wasserstein Geometry Of Gaussian Measures" by Asuka Takatsu the author shows how the space of d-dimensional Gaussian probability measures with non-singular covariance ...
- 71
8
votes
1
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261
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Mass transportation proof of the Gaussian isoperimetric inequality?
In his book "Topics in optimal transportation", Graduate Studies in Mathematics 58, AMS 2003,
Villani presents a proof, due to Gromov, of the classical isoperimetric inequality
in Euclidean ...
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76
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Wasserstein compactness of sublevel sets of relative entropy
Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathcal{P}(\...
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$L^2$ metric on $\textrm{Diff}(M)$ and geodesics
The paper Geometry of diffeomorphism groups, complete integrability and optimal transport mentions the following:
The group $\textrm{Diff}(M)$ carries a natural $L^2$-metric
$\displaystyle \langle\...
- 185
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1
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For diffeomorphism $f$, if $X$ and $f(X)$ are both Gaussian, then $f$ is affine
I am trying to prove the following.
Let $f:\mathbb{R}^{n}\to \mathbb{R}^n$ be a diffeomorphism. If $X$ and
$f(X)$ are both $n$ -dimensional Gaussian variables, then $f$ is
affine. That is, there ...
- 185
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47
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Lipschitzness of conditional law of a stochastic filtering problem wrt the Wasserstein distance
Let $(X_t)_{t\ge 0}$ and $(Y_t)_{t\ge 0}$ be a pair of stochastic processes taking values in $\mathbb{R}^n$ and in $\mathbb{R}^m$; defined on a filtered probability spaces $(\Omega,\mathcal{F},(\...
1
vote
1
answer
71
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Strict convexity of the cost function is enough to ensure the existence and uniqueness of the optimal transport map
Let $X=Y = \mathbb R^d$ and $c:X \times Y \to [0, \infty)$ be Borel measurable. Let $\mu, \nu$ be Borel probability measures on $X,Y$ respectively. Let $\mathcal T$ be the set of all Borel measurable ...
- 713
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Optimal Transport: how is this transport map Borel measurable?
I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
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1
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Bounds on the Bures–Wasserstein distance
The Bures–Wasserstein distance between $n\times n$ positive semidefinite matrices $A$ and $H$ is defined to be
$$
d(A,H) := \left[ \operatorname{tr} A + \operatorname{tr} H - 2\operatorname{tr} (A^{1/...
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41
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Even transport map and Brenier map in optimal transport?
I am working on a topic related to even transport map, i.e. $T(-x) = T(x)$. It's known that if $T\# \mu =\nu$ where $\mu = f(x)dx$ and $\nu = g(y)dy$, then $T$ satisfies the Jacobian equation $g(T(x)) ...
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1
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Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?
Let $(E, d)$ be a Polish space and $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$. Let $p \in [1, \infty)$ and $\mathcal P_p (E)$ the space of all Borel probability ...
- 713
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1
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Optimal transport: the existence of an optimal pair of $c$-conjugate functions
$\newcommand{\diff}{ \, \mathrm d}$
Let
$X,Y$ be Polish spaces,
$\mathcal C_b(X)$ the space of all real-valued bounded continuous functions on $X$,
$\mathcal P(X)$ the space of Borel probability ...
- 713
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0
answers
113
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Optimal transport: how is the use of disintegration theorem valid in this construction of $\widetilde{\phi}$?
Let $X,Y$ be Polish spaces and $\mathcal P(X)$ the space of all Borel probability measures on $X$.
Fix $\mu\in \mathcal P(X), \nu \in \mathcal P(Y)$. Let $\pi \in \Pi(\mu, \nu)$, i.e., $\pi \in \...
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1
answer
112
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Optimal transport: how $\varphi^c$ can be written as $\varphi^c = \lim _{\ell \rightarrow \infty} \psi_{\ell}$?
Let $X,Y$ be Polish spaces and $c:X \times Y \to [0, \infty]$ lower semi-continuous. There is a sequence $(c_\ell)_{\ell \in \mathbb N}$ with $c_\ell:X \times Y \to [0, \infty)$ of bounded Lipschitz ...
- 713
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43
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$c$-cyclical monotonicity: does this proof hold if $f \equiv +\infty$ or $\int c \mathrm d \gamma = +\infty$?
I'm reading the proof of Theorem 1.38. from section 1.6.2 $c$-cyclical monotonicity and duality of Santambrogio's Optimal transport for applied mathematicians.
My understanding: It seems for the ...
- 713
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vote
1
answer
109
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Are convex functions on manifolds the same as $c$-convex functions, where $c(x,y)=d(x,y)^2/2$?
I am reading the following book on optimal transport. While reading I came across the following definition of $c-$convexity.
Given $X$ and $Y$ metric spaces, $c: X \times Y \rightarrow \mathbb{R}$, ...
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42
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"More" cyclical monotonicity
Let $X$ and $Y$ be some finite sets. For a given function $f:X\times Y\rightarrow \mathbb{R}$, we say a set $S\subset X\times Y$ is $f$-cyclically monotone if for any sequence $(x_1,y_1),...,(x_n,y_n)\...
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84
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Continuity equation for a density of a measure
From the paper of Ambrosio-Crippa, it is known that if $\beta:\mathbb R^d\times[0, T[\longrightarrow\mathbb R^d$ is suitably regular, then the system
$$
\begin{cases}
\dfrac{\partial\mu}{\partial t}(x,...
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answers
69
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Continuity of the traffic intensity
Note: We say that a Borel measure on $\mathbb R^n$ has a continuous density if it is absolutely continuous with respect to Lebesgue measure and has a continuous Radon Nikodym derivative.
Let $\mu$ be ...
- 2,574
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56
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Conditions for existence of an optimal transport map between finitely supported probability measures
Let $E$ be a polish and let $P$ and $Q$ be finitely supported probability measures on $X$. What conditions are required to ensure that: for every $\delta>0$ there exists a $\delta$-optimal ...
- 11
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1
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Distance between empirical measures and thickened version
Let $\mathcal{H}$ be a separable Hilbert space and let $x_1,...,x_n$ be points in $\mathcal{H}$. Let $\varepsilon >0 $ be given and consider the measures
$$
\mu := \frac1{n}\,\sum_{i=1}^n\, \...
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1
answer
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Approximation of two densities with a single transformation
Let $p_1$ and $p_2$ be two probability densities and $X_i\sim N(\mu_i,\Sigma_i)$. Write $w(X)\sim p$ if the law of the random variable $w(X)$ has a density equal to $p$. For general densities $p_i$, ...
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solution of equivalent problem Kantorovich for case squared distance function
We know that the Kantorovich duality when the cost function is the square Euclidean distance is equivalent to
$$
\inf_{(\tilde\varphi,\tilde\psi)\in \tilde\Phi_c} J(\tilde\varphi,\tilde\psi) = \sup_{\...
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What are the "applications" of quantum optimal transport?
A quantum version of the Monge-Kantorovich optimal transport problem aims at optimizing a Hermitian cost matrix $C$ over the set of all bipartite coupling states $\rho_{AB}$, s.t. both of its reduced ...
user478492
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1
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Optimal transport for applied mathematicians: how does $\varphi (x) = \inf_{y \in Y} [c(x, y) - \psi (y)] \neq -\infty$ follow in Theorem 1.37?
I'm reading a proof of Theorem 1.37 from Santambrogio's Optimal transport for applied mathematicians: calculus of variations, PDEs, and modeling. First, I quote related definitions. Let $X,Y$ be ...
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2
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148
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How to compute the unique disintegration w.r.t. the first coordinate?
Set $\pi=\frac{1}{4}(\delta_{(1,0)}++\delta_{(1,3)}+\delta_{(1,1)}+\delta_{(2,2)})$. Suppose that $\pi\in\Pi(\mu,\nu)$.
How to get the disintegration of $\pi$ with respect to $\mu$?
- 216
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64
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Does $\mathscr{H}^{d-1} (A)<+\infty$ for $A\subset \mathbb R^d$ imply $A$ is (Borel) measurable?
I'm reading section 1.3.1 The quadratic case in $\mathbb{R}^{d}$ at page 17 from Santambrogio's Optimal Transport for Applied Mathematicians. The PDF is freely available from here.
Let $\mu$ be a ...
- 713
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1
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Uniqueness of Kantorovich potentials?
$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth.
Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein ...
- 4,309
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0
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34
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Transport-type duality for preduals of $C^{k,1}$-functions
Let $\Omega$ be a non-empty, simply connected, and open subset of $\mathbb{R}^d$ for some positive integer $d$. Let $k$ be a non-negative integer. Consider the Banach space $C^{k,1}_0(\Omega)$ ...
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104
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Eigenvalues of Witten Laplacian induced by log-concave probability measure on manifold
Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\mu=e^{-V}d\mathrm{vol}_M$ be a log-concave probability measure on $M$, such that the pair $(M,\mu)$ verifies the so-called Bakry-Emery ...
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What is the role of of continuity in this proof of Kantorovich duality?
I'm reading the proof of Kantorovich duality from Villani's book Topics in Optimal Transportation.
Let $X$ and $Y$ be Polish spaces. Let $P(X), P(Y)$ be the spaces of all Borel probability measures ...
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1
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Why is $\Xi \equiv 0$ if $E=C_{0}(X \times Y)$?
I'm reading the proof of Kantorovich duality from Villani's book Topics in Optimal Transportation. In page 28, the author said that
Exercise 1.11. Let us try to extend this proof to the non-compact ...
- 713
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0
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Sobolev variant of Wasserstein space
Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...
- 4,881
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votes
1
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Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity?
In a smooth, bounded and convex domain $\Omega\subset \mathbb R^d$ consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions
\begin{equation}
\label{FP}
\...
- 4,309
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answers
222
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Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure
Let $(X, | \cdot |)$ be a Banach space.
I am interested in whether one can extend the definition of the Kullback-Leibler divergence
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...
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answer
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Given iid samples from the joint distribution $P$ of pair of r.v.'s $(X,Y)$, how to get iid samples from independence coupling $P_X \otimes P_Y$?
Let $(X,Y)$ be a pair of random variables on a measure space $\mathcal T \subseteq \text{"subsets of }\mathbb R^2\text{"}$, with joint probability distribution $P$.
We don't assume $X$ and $Y$ are ...
- 6,338
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votes
3
answers
298
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Hyperbolic space embeds into Wasserstein space
Fix a positive integer $n$, let $\mathbb{H}^n$ be the $n$-dimensional hyperbolic space, $r>0$, $x\in \mathbb{H}^n$ and consider the closed (compact) geodesic ball $B_{\mathbb{H}^n}(x,r)$. Are ...
- 115
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Hyperplanes which equalize the Radon transforms of two distributions
Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance ...
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1
vote
1
answer
51
views
Expectation of a function according to a family of distributions
Consider a family of smooth, atomless CDFs, $F_x(\cdot)$, for each $x \in \mathbb R$. Suppose that $F_x(\cdot)$ are FOSD ranked in $x$. That is, for any $x, x'$ such that $x \ge x'$, $F_x(\cdot) \le ...
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199
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Measurability of Markov kernel wrt the Borel $\sigma$-algebra generated by the weak topology
Consider two Polish metric probability spaces $(\mathcal{A}, \Sigma_\mathcal{A})$ and $(\mathcal{B}, \Sigma_\mathcal{B})$, endowed with their Borel $\sigma$-algebras. Denote as $\mathcal{P}_\mathcal{B}...
- 241
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0
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32
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Sufficient condition for an $n$-tuple to be a convex conjugate
We say $(f_1,f_2,\dotsc,f_N)$ is a convex conjugate if for any $i=1,2,\dotsc,N$ and any $x_i\in\Bbb R^d$, we have:
$$f_i(x_i)=\sup\left\{\sum_{k=1}^{N}\sum_{j=k+1}^N x_k x_j - \sum_{j=1,j\neq i}^N f_j(...
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2
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185
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Building the Wasserstein space by pushforwards
Let $\mathbb{R}^d$ denote the $d$-dimensional Euclidean space, $\mathcal{W}_2(\mathbb{R}^d)$ denote the $2$-Wasserstein space with respect to the $d$-dimensional Euclidean space $\mathbb{R}^d$. Let $...
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0
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75
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Estimate of Wasserstein distance and flow of vector fields under particular assumptions
Let $\mu$ be a compactly supported absolutely continuous probability measure. Let $v,u$ be Lipschitz vector fields. For a vector field $w$ recall that $\Phi_t^w$ denotes its flow.
A classical estimate ...
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