Fractional Sobolev processes on Wasserstein spaces and their energy-minimizing particle representations with applications

Given a probability-measure-valued process $(\mu_t)$, we aim to find, among all path-continuous stochastic processes whose one-dimensional time marginals coincide almost surely with $(\mu_t)$ (if there is any), a process that minimizes a given energy in expectation. Building on our recent study (arXiv:2502.12068), where the minimization of fractional Sobolev energy was investigated for deterministic paths on Wasserstein spaces, we now extend the results to the stochastic setting to address some applications that originally motivated our study. Two applications are given. We construct minimizing particle representations for processes on Wasserstein spaces on $\mathbb{R}$ with H\"{o}lder regularity, using optimal transportation. We prove the existence of minimizing particle representations for solutions to stochastic Fokker--Planck--Kolmogorov equations on $\mathbb{R}^\mathrm{d}$ satisfying an integrability condition, using the stochastic superposition principle of Lacker--Shkolnikov--Zhang (J. Eur. Math. Soc. 25, 3229--3288 (2023)).