A Priori Bounds for the $$\Phi ^4$$ Equation in the Full Sub-critical Regime

Abstract We derive a priori bounds for the $$\Phi ^4$$ ? 4 equation in full sub-critical regime using Hairer’s theory of regularity structures. The is formally given by "Equation missing"where term $$+\infty \phi $$ + ? ? represents infinite terms that have to be removed renormalisation procedure. emulate fractional dimensions $$d<4$$ d < adjusting noise $$\xi ? , choosing \in C^{-3+\delta }$$ ? C - 3 ? . Our main result states if $$\phi satisfies this on space–time cylinder $$D= (0,1) \times \{ |x| \leqslant 1 \}$$ D = ( 0 , 1 ) × { | x ? } then away from boundary $$\partial D$$ ? solution can bounded finite number explicit polynomial expressions bound holds uniformly over all possible choices data and thus relies crucially super-linear damping effect non-linear $$-\phi ^3$$ A key part our analysis consists an appropriate re-formulation structures specific context (*), which allows us couple small scale control one obtains with suitable large argument. Along way we make several new observations simplifications: reduce objects required respect work. Instead model $$(\Pi _x)_x$$ ? family translation operators $$(\Gamma _{x,y})_{x,y}$$ ? y work just single object $$(\mathbb {X}_{x, y})$$ X acts itself translations, very much spirit Gubinelli’s branched rough paths. Furthermore, show (*) hierarchy continuity conditions constitute definition modelled distribution reduced condition “coefficient constant level”.