A priori bounds for the $Φ^4$ equation in the full sub-critical regime

Abstract

We derive a priori bounds for the Φ 4 equation in the full subcritical regime using Hairer's theory of regularity structures. The equation is formally given by (∂t − ∆)φ = −φ 3 + ∞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 −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 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".

A diagram-free approach to the stochastic estimates in regularity structures

“…The use of a SG inequality requires the control of the (first-order) Malliavin derivative of Π − , which is the Fréchet derivative of Π − = Π − [ξ ] w. r. t. the noise ξ . It is convenient to think of it in terms of the directional derivative δ Π − for 9 this particularly simple form relies on the assumption that ξ is invariant in law under spatial reflection 10 in view of the stationarity of the noise 11 here and in the sequel, [•] denotes the functional dependence on a field 12 however, our approach is oblivious to symmetries arising from a Gaussian nature of the noise 13 i. e. stationarity 14 in the spatial variable(s) some arbitrary infinitesimal noise perturbation 15 δ ξ . Since Π − is multi-linear in ξ , passing to δ Π − amounts to replacing one of the instances of ξ by δ ξ .…”

“…As a collateral damage, in order to avoid critical cases, we have to generalize from white noise to a more general noise ξ with a fractional (negative) Sobolev norm playing the role of the Cameron-Martin space. Like in [9], this has the positive side effect of allowing to explore the limits of the approach. In order not to break scaling, we work with annealed instead of quenched estimates.…”

On the unique ergodicity for a class of 2 dimensional stochastic wave equations

“…A result about existence (and uniqueness) of an invariant measure for (1.4) would provide a new construction of the Φ 4 -measure, completing the project started by Parisi and Wu in their influential work [21]. This has recently been achieved on the torus in (fractional) dimension d < 4 in the papers [18,27,19,6], that show the relevant estimates to ensure the existence of an invariant measure via a Krylov-Bogolyubov argument. Together with the abstract ergodicity result of [15], this also allows to show an analogous version of Theorem 1.1 for the equation (1.4).…”

“…Following the approach suggested by [6], where γ = C d is an appropriate constant, the equation (1.1) corresponds to (1.5) for fractional dimension d = 2 − s. Equations of the form of (1.1) have already been considered in the literature by Barbu and Da Prato in [1], and Brzeźniak, Ondreját and Seidler in [3]. However, when applying the results of [1,3] to (1.1), we obtain Theorem 1.1 only for s > 1, which would correspond to an ergodicity result for (1.5) in fractional dimensions d < 1.…”

Flow equation approach to singular stochastic PDEs

“…However, its generalization for non-local singular SPDEs treated in the present work is non-trivial. Let us also mention that there are a number of results about specific families of singular SPDEs covering whole sub-critical region [9,10,37,42].…”