  When propagating uncertainty in the data of differential equations, the
probability laws describing the uncertainty are typically themselves subject to
uncertainty. We present a sensitivity analysis of uncertainty propagation for
differential equations with random inputs to perturbations of the input
measures. We focus on the elliptic diffusion equation with random coefficient
and source term, for which the probability measure of the solution random field
is shown to be Lipschitz-continuous in both total variation and Wasserstein
distance. The result generalizes to the solution map of any differential
equation with locally H\"older dependence on input parameters. In addition,
these results extend to Lipschitz continuous quantities of interest of the
solution as well as to coherent risk functionals of these applied to evaluate
the impact of their uncertainty. Our analysis is based on the sensitivity of
risk functionals and pushforward measures for locally H\"older mappings with
respect to the Wasserstein distance of perturbed input distributions. The
established results are applied, in particular, to the case of lognormal
diffusion and the truncation of series representations of input random fields.
