In this talk I will present a new method to derive a priori estimates for singular SPDE concentrating on the dynamic  Φ⁴₃ equation.
Recently several methods to show non-explosion for this equation in the framework of paracontrolled distributions were put forward by several groups. Here I will show how to prove bounds in the framework of regularity structures. The main result is a space-time version of the coming down from infinity property, i.e. a bound on solutions of the equation on a compact set in space-time which only depends on the stochastic data on a slightly larger set, but is uniform over all possible choices of space-time boundary data.
Our method significantly simplifies previous proofs and extensions to (much) more singular equations seem within reach.
This is joint work with Augustin Moinat.