The Dynamic $${\Phi^4_3}$$ Φ 3 4 Model Comes Down from Infinity

Abstract

Abstract:We prove an a priori bound for the dynamic 4 3 model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows one to construct invariant measures via the Krylov-Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean 4 3 field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities.

The invariant measure and the flow associated to the $\Phi^4_3$-quantum field model

“…The extension from local to global solutions in the case of T 3 is discussed in [53], and in [73] by an interplay of the paracontrolled approach in [50] with Bourgain's method, exploiting the presence of the candidate for an invariant measure, namely the weak coupling case Φ 4 3 -measure as discussed in [25]. It is asserted in the abstract of [73] that the existence of invariant measures follows from the proven uniform bounds on solutions "via the Krylov-Bogoliubov method" (details are not given in the paper). For the relation of such invariant measures with "the Φ 4 3 -measure" of quantum field theory see [57] and [73].…”

“…It is asserted in the abstract of [73] that the existence of invariant measures follows from the proven uniform bounds on solutions "via the Krylov-Bogoliubov method" (details are not given in the paper). For the relation of such invariant measures with "the Φ 4 3 -measure" of quantum field theory see [57] and [73].…”

The KPZ equation on the real line

“…First steps to overcome this restriction were taken by Hairer and Labbé [HL15,HL18] in their study of the linear rough heat equation and the linear parabolic Anderson model on the whole Euclidean space. For non-linear equations, a priori estimates are a natural and powerful tool and they were very successful in the study of the Φ 4 d equations in the work by Mourrat and Weber [MW17a,MW17b], by Gubinelli and Hofmanova [GH18b], and by Barashkov and Gubinelli [BG18], all relying on the damping induced by the term −φ 3 . But such estimates depend strongly on the structure of the equation and this prevents the development of a general solution theory for singular SPDEs in infinite volume.…”

Local bounds for stochastic reaction diffusion equations

“…The theory of regularity structures is indeed a main motivation for this work. A priori including the "coming-down from infinitiy" property have been proven for singular SPDEs, namely the dynamic φ 2m 2 [16,20] and φ 4 3 models [15,2,9] both on compact domains and on the full space. The works on φ 4 3 all relied on Fourier methods, the method of paracontrolled distributions, rather than the theory of regularity structures.…”