Alberti's type rank one theorem for martingales

TL;DR

This paper extends Alberti's rank-one theorem to martingales by analyzing the polar decomposition of vector measures' singular parts through conditional expectations and a martingale wave cone, linking PDE measure properties.

Contribution

It introduces a martingale version of Alberti's rank-one theorem, connecting measure decomposition with martingale wave cones and PDE-constrained measure properties.

Findings

Established a martingale analog of Alberti's rank-one theorem.

Linked the polar decomposition of vector measures to martingale wave cones.

Extended PDE measure properties to the martingale setting.

Abstract

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.