Wasserstein metric and subordination

Philippe Clément ; Wolfgang Desch

Studia Mathematica, Tome 187 (2008), p. 35-52 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

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Publié le : 2008-01-01

Zbl 1152.60010

EUDML-ID : urn:eudml:doc:285264