Sensitivity of Uncertainty Propagation for the Elliptic Diffusion Equation
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For elliptic diffusion equations with random coefficient and source term, the probability measure of the solution random field is shown to be Lipschitz-continuous in both total variation and Wasserstein distance as a function of the input probability measure. These results extend to Lipschitz continuous quantities of interest of the solution as well as to coherent risk functionals of those applied to evaluate their uncertainty. Our analysis is based on the sensitivity of risk functionals and pushforward measures for locally Lipschitz mappings with respect to the Wasserstein distance of perturbed input distributions. The established results particularly apply to the case of lognormal diffusions and the truncation of series representations of input random fields.
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