Functional uniqueness and stability of Gaussian priors in optimal L1 estimation

This paper studies the functional uniqueness and stability of Gaussian priors in optimal $L^1$ estimation. While it is well known that the Gaussian prior uniquely induces linear conditional means under Gaussian noise, the analogous question for the conditional median (i.e., the optimal estimator under absolute-error loss) has only recently been settled. Building on the prior work establishing this uniqueness, we develop a quantitative stability theory that characterizes how approximate linearity of the optimal estimator constrains the prior distribution. For $L^2$ loss, we derive explicit rates showing that near-linearity of the conditional mean implies proximity of the prior to the Gaussian in the Lévy metric. For $L^1$ loss, we introduce a Hermite expansion framework and analyze the adjoint of the linearity-defining operator to show that the Gaussian remains the unique stable solution. Together, these results provide a more complete functional-analytic understanding of linearity and stability in Bayesian estimation under Gaussian noise.