Journal
MATHEMATISCHE NACHRICHTEN
Volume 288, Issue 17-18, Pages 2028-2041Publisher
WILEY-V C H VERLAG GMBH
DOI: 10.1002/mana.201400408
Keywords
Wasserstein metric; self-similar measure; self-similar coupling; Bernoulli convolution
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Funding
- EPSRC grant [EP/J013560/1]
- Engineering and Physical Sciences Research Council [EP/J013560/1] Funding Source: researchfish
- EPSRC [EP/J013560/1] Funding Source: UKRI
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We study aspects of the Wasserstein distance in the context of self-similar measures. Computing this distance between two measures involves minimising certain moment integrals over the space of couplings, which are measures on the product space with the original measures as prescribed marginals. We focus our attention on self-similar measures associated to equicontractive iterated function systems consisting of two maps on the unit interval and satisfying the open set condition. We are particularly interested in understanding the restricted family of self-similar couplings and our main achievement is the explicit computation of the 1st and 2nd moment integrals for such couplings. We show that this family is enough to yield an explicit formula for the 1st Wasserstein distance and provide non-trivial upper and lower bounds for the 2nd Wasserstein distance for these self-similar measures. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhetm
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