A priori bounds for the φ⁴ equation in the full sub-critical regime

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 (∂t − )φ = −φ3 + ∞φ + ξ , 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 −ϕ3 . 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 (Xx,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”.