Second-order Wasserstein response for Levy laws

Two perturbations can have the same size in a weak metric and still produce terminal Wasserstein errors of different orders. We study this distinction for Gaussian-smoothed infinitely divisible laws by differentiating their distribution functions with respect to variance-weighted Lévy characteristics. The framework includes singular directions generated by moving atoms and gives first- and second-order expansions with remainders uniform over Lipschitz tests. When the first response is nonzero, Gaussian analyticity identifies the signed second-order coefficient in the Wasserstein–1 expansion. When the first tangent vanishes, a finite second displacement moment is enough to obtain a quadratic response, with a strictly positive coefficient whenever the displacement dispersion is nonzero. We apply these formulas to local balanced remeshing and show that the error is comparable to the grid-alignment variance. This establishes the sharpness of the quadratic grid rate for compactly supported densities and nonaligned atomic sequences. We also derive metric speed and length along admissible non-atomic curves and formulate a response-based linear program, with a certified oracle gap and consistent spatial quadrature. Finally, we describe the sharp transition as Gaussian smoothing vanishes and extend the second-order analysis to several dimensions. The Supplement contains the state-dependent response formulas and further stability estimates.