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Upon its inception the theory of regularity structures allowed for thetreatment for many semilinear perturbations of the stochastic heat equationdriven by space-time white noise. When the driving noise is non-Gaussian themachinery of theory can still be used but must be combined with an infinitenumber of stochastic estimates in order to compensate for the loss ofhypercontractivity. In this paper we obtain a more streamlined and automaticset of criteria implying these estimates which facilitates the treatment ofsome other problems including non-Gaussian noise such as some general phasecoexistence models - as an example we prove here a generalization of theWong-Zakai Theorem found by Hairer and Pardoux.

Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem