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Wednesday, July 9, 2025 - 14:00 in v3-201+Zoom


Energy-minimizing particle representations for low-regularity paths on Wasserstein spaces

A talk in the Bielefeld Stochastic Afternoon series by
Ehsan Abedi

Abstract: We study a generalization of the Monge--Kantorovich optimal transportation problem. Given a prescribed family of time-dependent probability measures $(\mu_t)$, we aim to find, among all stochastic processes on a suitable path space whose one-dimensional time marginals coincide with $(\mu_t)$ (if there is any), a process that minimizes a given energy. Starting from existing results, we highlight a superposition principle based on optimal transport, which, in particular, yields Sobolev-energy-minimizing processes for absolutely continuous curves $(\mu_t)$ in $p$-Wasserstein spaces with $p > 1$. We then report our recent progress. First, we discuss the special case of $p=1$, and more generally, c\`adl\`ag curves of bounded variation in $1$-Wasserstein spaces. Second, we consider fractional Sobolev paths (resp. processes) in $p$-Wasserstein spaces (resp. of random measures). As an application, we prove the existence of fractional Sobolev-energy-minimizing processes for solutions to stochastic Fokker--Planck equations, using the stochastic superposition principle by Lacker--Shkolnikov--Zhang. The talk is based on two preprints (arXiv:2502.12068, 2503.10859) and joint work with Zhenhao Li and Timo Schultz (Calc.Var.63,16 (2024)).

Within the CRC this talk is associated to the project(s): A5



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