First and second moments for self-similar couplings and Wasserstein distances

Abstract

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 satisfying the open set condition and consisting of two maps on the unit interval. 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.Mathematics Subject Classification 2010: Primary: 28A80, 28A33, 60B05. Secondary: 28A78.

Stationary measures associated to analytic iterated function schemes

“…This answers in affirmative a conjecture of J. Fraser, from Section 4 of . Corollary 2.6 of that paper contains the special case of Theorem when the contractions rates of T 1 and T 2 are the same (i.e.…”

“…Firstly we give a simple proof of the fact that the integrals of analytic functions with respect to the stationary measure vary analytically if we perturb the contractions and the weights analytically. Secondly, we consider the special case of affine contractions and we prove a conjecture of J. Fraser in on the Kantorovich–Wasserstein distance between two stationary measures associated to affine contractions on the unit interval with different rates of contraction.…”

Average distances on self-similar sets and higher order average distances of self-similar measures

“…It is well-known that convergence with respect to the Kantorovich-Wasserstein distance is equivalent to weak convergence. The Kantorovich-Wasserstein distance between self-similar measures has recently been studied by Fraser [11] and investigated further by Cipriano and Pollicott [4] and Cipriano [5]. In particular, Fraser [11] found an explicit formula for the Kantorovich-Wasserstein distance between two self-similar measures on the real line generated by Iterated Function Systems of two maps with a common contractions ratio.…”

“…The Kantorovich-Wasserstein distance between self-similar measures has recently been studied by Fraser [11] and investigated further by Cipriano and Pollicott [4] and Cipriano [5]. In particular, Fraser [11] found an explicit formula for the Kantorovich-Wasserstein distance between two self-similar measures on the real line generated by Iterated Function Systems of two maps with a common contractions ratio. For the self-similar measures μ p and μ q , the product measure μ p × μ q is clearly a coupling of μ p and μ q (but typically not a coupling that realises the Kantorovich-Wasserstein distance between μ p and μ q ), and our results are therefore related to the study of the Kantorovich-Wasserstein distance between self-similar measures in [4,5,11].…”

“…In particular, Fraser [11] found an explicit formula for the Kantorovich-Wasserstein distance between two self-similar measures on the real line generated by Iterated Function Systems of two maps with a common contractions ratio. For the self-similar measures μ p and μ q , the product measure μ p × μ q is clearly a coupling of μ p and μ q (but typically not a coupling that realises the Kantorovich-Wasserstein distance between μ p and μ q ), and our results are therefore related to the study of the Kantorovich-Wasserstein distance between self-similar measures in [4,5,11]. For example, in order to derive his main results, Fraser [11, Theorem 2.1] first finds an explicit formula for the average |x − y| dγ r (x, y) for a certain family of self-similar measures γ r indexed by a parameter r , and a special case of this result is a special case of Theorem 2.1 providing an explicit formula for the average |x − y| d(μ p × μ q )(x, y).…”