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

Abstract We derive a priori bounds for the ⁴ Φ 4 equation in the full sub-critical regime using Hairer’s theory of regularity structures. The equation is formally given by "Equation missing"where the term + + ∞ ϕ represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions d d 4 by adjusting the regularity of the noise term ξ, choosing C^-3+ ξ ∈ C - 3 + δ. Our main result states that if ϕ satisfies this equation on a space–time cylinder D= (0, 1) \ |x| 1 \ D = (0, 1) × | x | ⩽ 1, then away from the boundary D ∂ 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 - ³ - 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 (ₓ) ₓ (Π x) x and the family of translation operators (ₗ, ₘ) ₗ, ₘ (Γ x, y) x, y we work with just a single object (Xₗ, ₘ) (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”.