Local expansion properties of paracontrolled systems

In recent years, the theories of regularity structures and paracontrolled calculus enabled the study of singular SPDE, providing the appropriate functional framework to handle renormalisation of these equations. While regularity structures uses local descriptions of distributions through generalized Taylor expansions,  paracontrolled calculus takes a more global approach, utilizing harmonic analysis and the concept of paracontrolled systems. We will present a result that connects these two theories by introducing a family of regularity structures that captures the local behavior of paracontrolled systems.