Corrigendum to "Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions"

TL;DR

This corrigendum addresses a mistake in a previous paper on FitzHugh-Nagumo SPDEs, providing a corrected proof that extends the results to full generality using a modified fixed-point formulation.

Contribution

The paper introduces an alternative formulation of the fixed-point problem with a modified integration operator to correct and generalize earlier results.

Findings

Corrected regularity estimates for FitzHugh-Nagumo SPDEs.

Extended validity of key theorems to all space dimensions and nonlinearities.

Developed new multilevel Schauder estimates and renormalisation analysis.

Abstract

Lemma 4.8 in the article [Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions, Electronic J. Probability 21 (18):1-48 (2016), arXiv:1504.02953] contains a mistake, which implies a weaker regularity estimate than the one stated in Proposition 4.11. This does not affect the proof of Theorem 2.1, but Theorems 2.2 and 2.3 only follow from the given proof if either the space dimension $d$ is equal to $2$, or the nonlinearity $F(U,V)$ is linear in $V$. To fix this problem and provide a proof of Theorems 2.2 and 2.3 valid in full generality, we consider an alternative formulation of the fixed-point problem, involving a modified integration operator with nonlocal singularity and a slightly different regularity structure. We provide the multilevel Schauder estimates and renormalisation-group analysis required for the fixed-point argument in this new setting.