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Alberti's type rank one theorem for martingales
Rami Ayoush, Dmitriy Stolyarov, Michał Wojciechowski · 2023-07-21 · via math.PR updates on arXiv.org

We prove that the polar decomposition of the singular part of a vector measure depends on its conditional expectations computed with respect to the $q$-regular filtration. This dependency is governed by a martingale analog of the so-called wave cone, which naturally corresponds to the result of De Philippis and Rindler about fine properties of PDE-constrained vector measures. As a corollary we obtain a martingale version of Alberti's rank-one theorem.

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