Abstract
We prove a sharp general inequality estimating the distance of two probability measures on a compact Lie group in the Wasserstein metric in terms of their Fourier transforms. We use a generalized form of the Wasserstein metric, related by Kantorovich duality to the family of functions with an arbitrarily prescribed modulus of continuity. The proof is based on smoothing with a suitable kernel, and a Fourier decay estimate for continuous functions. As a corollary, we show that the rate of convergence of random walks on semisimple groups in the Wasserstein metric is necessarily almost exponential, even without assuming a spectral gap. Applications to equidistribution and empirical measures are also given.
Original language | English |
---|---|
Article number | 13 |
Number of pages | 23 |
Journal | The Journal of Fourier Analysis and Applications |
Volume | 27 |
Issue number | 2 |
DOIs | |
Publication status | Published - Apr 2021 |
Keywords
- Compact group
- Equidistribution
- Erdős–Turán inequality
- Fourier transform
- Random walk
- Spectral gap
- Transport metric
ASJC Scopus subject areas
- Analysis
- Applied Mathematics
- Mathematics(all)