A priori bounds for quasi-linear SPDEs in the full subcritical regime

This paper is concerned with quasi-linear parabolic equations driven by an additive forcing C^-2, in the full subcritical regime (0, 1). We are inspired by Hairer’s regularity structures, however we work with a more parsimonious model indexed by multi-indices rather than trees. This allows us to capture additional symmetries which play a crucial role in our analysis. Assuming bounds on this model, which is modified in agreement with the concept of algebraic renormalization, we prove local a priori estimates on solutions to the quasi-linear equations modified by the corresponding counter-terms.