A Priori Bounds for the Φ4 Equation in the Full Sub-critical Regime

Chandra, Ajay; Moinat, Augustin; Weber, Hendrik

Abstract

We derive a priori bounds for the Φ4 equation in the full sub-critical regime using Hairer’s theory of regularity structures. The equation is formally given by where the term represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions by adjusting the regularity of the noise term , choosing . Our main result states that if satisfies this equation on a space–time cylinder , then away from the boundary the solution can be bounded in terms of a finite number of explicit polynomial expressions in . The bound holds uniformly over all possible choices of boundary data for and thus relies crucially on the super-linear damping effect of the non-linear term . A key part of our analysis consists of an appropriate re-formulation of the theory of regularity structures in the specific context of (*), which allows us to couple the small scale control one obtains from this theory with a suitable large scale argument. Along the way we make several new observations and simplifications: we reduce the number of objects required with respect to Hairer’s work. Instead of a model and the family of translation operators we work with just a single object which acts on itself for translations, very much in the spirit of Gubinelli’s theory of branched rough paths. Furthermore, we show that in the specific context of (*) the hierarchy of continuity conditions which constitute Hairer’s definition of a modelled distribution can be reduced to the single continuity condition on the “coefficient on the constant level”.