Renormalisation in the flow approach for singular SPDEs
Yvain Bruned, Aurélien Minguella
TL;DR
This work develops a bottom-up, decorated-tree framework for renormalising singular SPDEs within Duch's flow approach and proves that the resulting renormalisation coincides with the classical BPHZ renormalisation from regularity structures. It introduces a recursive evaluation map $\Pi_{\varepsilon,\mu}^R$ built from a localisation map $M_{\text{loc}}$ and elementary differentials $\Upsilon$, together with a root-extraction coproduct $\Delta_r$ and a derivative $\uparrow_\mu$, yielding a Polchinski-type flow on the coefficients. The main results establish coherence (the ansatz satisfies the flow up to a projector), derive the renormalised SPDE, and show that a localization of the evaluation map reproduces the BPHZ scheme by selecting $\ell_{\text{BPHZ}}$. This work thereby clarifies the algebraic underpinnings and combinatorics of renormalisation in the flow framework and provides a rigorous bridge to regularity-structures renormalisation for subcritical singular SPDEs.
Abstract
In this work, we study the renormalisation of singular SPDEs in the flow approach recently developed by Duch using a bottom-up setting. We introduce a general ansatz based on decorated trees for the solution of the flow equation. The ansatz is renormalised in a recursive way, in the sense of the trees, via local extractions introduced for regularity structures. We derive the renormalised equation from this ansatz and show that the renormalisation scheme is identical to that appearing in the context of regularity structures, thus matching the BPHZ renormalisation.