A Priori Bounds for the Φ ⁴ Equation in the Full Sub-critical Regime

We derive a priori bounds for the Φ ⁴ equation in the full sub-critical regime using Hairer’s theory of regularity structures. The equation is formally given by [Equation not available: see fulltext.]where the term + ∞ϕ represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions d< 4 by adjusting the regularity of the noise term ξ , choosing ξ∈ C ^{- 3 + δ} . Our main result states that if ϕ satisfies this equation on a space–time cylinder D= (0 , 1) × { | x| ⩽ 1 } , then away from the boundary ∂D the solution ϕ can be bounded in terms of a finite number of explicit polynomial expressions in ξ . The bound holds uniformly over all possible choices of boundary data for ϕ and thus relies crucially on the super-linear damping effect of the non-linear term - ϕ ³ . 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 (Πx)x and the family of translation operators (Γx,y)x,y we work with just a single object (X _{x , y}) 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”.