Second-order Wasserstein response for Lévy laws
Abstract
Weak perturbations of the same size can lead to terminal Wasserstein errors of different orders. For infinitely divisible laws smoothed by a Gaussian, we differentiate their distribution functions with respect to variance-weighted Lévy characteristics. This includes distributional directions caused by moving atoms. We obtain first- and second-order expansions that are uniform over Lipschitz test functions. When the first response is nonzero, Gaussian analyticity gives the signed second coefficient of the Wasserstein--1 distance. When the first response vanishes, a finite second displacement moment is enough to obtain a quadratic response. Its coefficient is strictly positive if the displacement dispersion is nonzero. For local balanced remeshing, the error is comparable to the grid-alignment variance. This gives sharp quadratic grid rates for compactly supported densities and nonaligned atomic sequences. We derive metric speed and length along admissible non-atomic curves, and use the response to build a linear program with a proved bound on how much its error exceeds the infimum over the allowed class and consistent quadrature. We also prove a sharp transition as smoothing vanishes and give a multivariate second-order expansion. The appendix gives state-dependent response formulas and further stability estimates.
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BibTeX
@misc{autran2026secondorder,
title = {{Second-order Wasserstein response for Lévy laws}},
author = {Alexandre Autran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01411},
eprint = {2610.01411},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01411},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Second-order Wasserstein response for Lévy laws
AU - Alexandre Autran
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.01411
UR - https://arxiv.org/abs/2610.01411
AB - Weak perturbations of the same size can lead to terminal Wasserstein errors of different orders. For infinitely divisible laws smoothed by a Gaussian, we differentiate their distribution functions with respect to variance-weighted Lévy characteristics. This includes distributional directions caused by moving atoms. We obtain first- and second-order expansions that are uniform over Lipschitz test functions. When the first response is nonzero, Gaussian analyticity gives the signed second coefficient of the Wasserstein--1 distance. When the first response vanishes, a finite second displacement moment is enough to obtain a quadratic response. Its coefficient is strictly positive if the displacement dispersion is nonzero. For local balanced remeshing, the error is comparable to the grid-alignment variance. This gives sharp quadratic grid rates for compactly supported densities and nonaligned atomic sequences. We derive metric speed and length along admissible non-atomic curves, and use the response to build a linear program with a proved bound on how much its error exceeds the infimum over the allowed class and consistent quadrature. We also prove a sharp transition as smoothing vanishes and give a multivariate second-order expansion. The appendix gives state-dependent response formulas and further stability estimates.
ER -