In this talk, we examine the Monge and Beckmann formulations of optimal transport in infinite-dimensional settings. While in finite dimensions these problems are known to be equivalent, extending the theory to infinite-dimensional separable Banach spaces introduces fundamental structural obstacles: the absence of a canonical reference measure, the presence of singular shift directions, non-strict convexity of the cost, and the failure of coercivity in standard function spaces.

After reviewing these general limitations, we specialize to the setting of centered Gaussian measures, where the Cameron--Martin space provides a natural tangent structure and the divergence constraint is governed by the extended stochastic integral.

In this framework, we contrast two prominent methodologies: the analytical approach (as utilized by Riabov), which employs the Ornstein--Uhlenbeck semigroup as a regularizing mollifier to obtain a Beckmann-type variational representation for the $W_1$ distance, and the geometrical approach (stemming from Sudakov's ray-disintegration and developed on Wiener spaces by Cavalletti), which constructs optimal transport maps via measure disintegration along transport rays. Finally, we illustrate the subtleties of these frameworks through concrete examples.

Bio: Andrii Bobrov is a Master's student in mathematics at Kyiv Academic University, working under the supervision of Georgii Riabov. His research focuses on the L1-optimal transport and its geometric and functional properties.