Joint stochastic localization and applications

Stochastic localization is a pathwise analysis technique that has emerged as a powerful tool in high-dimensional probability and sampling. In this work, we extend stochastic localization to a joint framework for coupling probability measures and explore its applications in distributional data analysis. We first unify existing stochastic localization processes under Eldan's α-scheme and characterize their localization rates. Building on this, we introduce a joint scheme to couple probability measures via concurrent α-schemes driven by a shared Brownian motion. This construction is canonical and induces a family of metrics on the space of probability measures, which we call Eldan's α-distance. Alternative variants that extrapolate optimal Gaussian couplings to log-concave measures are also discussed. We study the theoretical properties of Eldan's α-distance, including its restriction to Gaussian measures and its behavior under affine transformations. For α = 0, we show it is topologically equivalent to the 2-Wasserstein distance for measures supported on a common compact set; we also relate its weighted variants to linearized optimal transport in Wiener space and to score-matching objectives in training diffusion models. Computationally, we develop efficient estimators for Eldan's α-distance in the cases α=0 and α=1/2, with rigorous error guarantees for log-concave and finitely supported measures in the former setting and Gaussian measures in the latter. Finally, we apply Eldan's α-distance as a scalable surrogate for the 2-Wasserstein distance to enable fast pairwise distance estimation and approximate computation of Wasserstein barycenters.