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We provide a relatively compact proof of the BPHZ theorem for regularitystructures of decorated trees in the case where the driving noise satisfies asuitable spectral gap property, as in the Gaussian case. This is inspired bythe recent work [LOTT21] in the multi-index setting, but our proof reliescrucially on a novel version of the reconstruction theorem for a space of"pointed Besov modelled distributions". As a consequence, the analytical coreof the proof is quite short and self-contained, which should make it easier toadapt the proof to different contexts (such as the setting of discrete models).

The BPHZ Theorem for Regularity Structures via the Spectral Gap Inequality