Wasserstein metric and subordination

Philippe Clément, and Wolfgang Desch. "Wasserstein metric and subordination." Studia Mathematica 189.1 (2008): 35-52. <http://eudml.org/doc/285264>.

@article{PhilippeClément2008,
abstract = {Let $(X,d_X)$, $(Ω,d_\{Ω\})$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_\{p\}$ the p-Wasserstein metric with some p ∈ [1,∞), and by $_\{p\}(X)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f: Ω → _\{p\}(X)$, $ω ↦ f_\{ω\}$, be a Borel map such that f is a contraction from $(Ω,d_\{Ω\})$ into $(_\{p\}(X),W_\{p\})$. Let ν₁,ν₂ be probability measures on Ω with $W_\{p\}(ν₁,ν₂)$ finite. On X we consider the subordinated measures $μ_\{i\} = ∫_\{Ω\} f_\{ω\}dν_\{i\}(ω)$. Then $W_\{p\}(μ₁,μ₂) ≤ W_\{p\}(ν₁,ν₂)$. As an application we show that the solution measures $ϱ_\{α\}(t)$ to the partial differential equation $∂/∂t ϱ_\{α\}(t) = -(-Δ)^\{α/2\}ϱ_\{α\}(t)$, $ϱ_\{α\}(0) = δ₀$ (the Dirac measure at 0), depend absolutely continuously on t with respect to the Wasserstein metric $W_\{p\}$ whenever 1 ≤ p < α < 2.},
author = {Philippe Clément, Wolfgang Desch},
journal = {Studia Mathematica},
keywords = {Wasserstein metric; probability measures on metric spaces; subordination},
language = {eng},
number = {1},
pages = {35-52},
title = {Wasserstein metric and subordination},
url = {http://eudml.org/doc/285264},
volume = {189},
year = {2008},
}

TY - JOUR
AU - Philippe Clément
AU - Wolfgang Desch
TI - Wasserstein metric and subordination
JO - Studia Mathematica
PY - 2008
VL - 189
IS - 1
SP - 35
EP - 52
AB - Let $(X,d_X)$, $(Ω,d_{Ω})$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_{p}$ the p-Wasserstein metric with some p ∈ [1,∞), and by $_{p}(X)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f: Ω → _{p}(X)$, $ω ↦ f_{ω}$, be a Borel map such that f is a contraction from $(Ω,d_{Ω})$ into $(_{p}(X),W_{p})$. Let ν₁,ν₂ be probability measures on Ω with $W_{p}(ν₁,ν₂)$ finite. On X we consider the subordinated measures $μ_{i} = ∫_{Ω} f_{ω}dν_{i}(ω)$. Then $W_{p}(μ₁,μ₂) ≤ W_{p}(ν₁,ν₂)$. As an application we show that the solution measures $ϱ_{α}(t)$ to the partial differential equation $∂/∂t ϱ_{α}(t) = -(-Δ)^{α/2}ϱ_{α}(t)$, $ϱ_{α}(0) = δ₀$ (the Dirac measure at 0), depend absolutely continuously on t with respect to the Wasserstein metric $W_{p}$ whenever 1 ≤ p < α < 2.
LA - eng
KW - Wasserstein metric; probability measures on metric spaces; subordination
UR - http://eudml.org/doc/285264
ER -