A priori bounds for the generalized parabolic Anderson model in the full subcritical regime

We show a priori bounds for the generalized Parabolic Anderson Model $({\mathrm{\partial }}_t-\mathrm{\Delta })u=\sigma (u)\diamond \xi$ in finite volume in the full subcritical regime, using the framework of regularity structures. The approach is based on the observation that for constant initial data the guaranteed existence time is independent of its size, followed by a specific nonlinear transformation to formulate a PDE for an auxiliary field $v$ which follows a pure transport type equation and which stays uniformly close to $u$. In particular, there are no further algebraic arguments involved which are not already present in the formulation of the local theory.