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 |