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

Abstract

We give a direct construction of invariant measures and global flows for the stochastic quantization equation to the quantum field theoretical Φ 4 3 -model on the 3-dimensional torus. This stochastic equation belongs to a class of singular stochastic partial differential equations (SPDEs) presently intensively studied, especially after Hairer's groundbreaking work on regularity structures. Our direct construction exhibits invariant measures and flows as limits of the (unique) invariant measures for corresponding finite dimensional approximation equations. Our work is done in the setting of distributional Besov spaces, adapting semigroup techniques for solving nonlinear dissipative parabolic equations on such spaces and using methods that originated from work by Gubinelli et al on paracontrolled distributions for singular SPDEs.

Some recent progress in singular stochastic partial differential equations

“…Then in [GH18b, GH18a] established a priori estimates for solutions on the full space R 3 yielding the construction of Φ 4 quantum field theory on the whole R 3 , as well as verification of some key properties that this invariant measure must satisfy as desired by physicists such as reflection positivity. See also [AK17]. Similar uniform a priori estimates are obtained by [MW18] using maximum principle.…”

Local bounds for stochastic reaction diffusion equations

“…This is the content of our companion paper [14]. There we show that our method significantly simplifes the technical arguments used in [15,2,9] and extend its scope to construct solutions on the full space without the need for weights.…”

“…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.…”

On the Probabilistic Cauchy Theory for Nonlinear Dispersive PDEs

“…In the context of nonlinear dispersive PDEs, probabilistic construction of solutions was initiated in an effort to construct well-defined dynamics almost surely with respect to the Gibbs measure for NLS on T d , d = 1, 2 [9,54,10]. Before discussing this problem for NLS on T d , let us consider the following finite dimensional 2 In fact, there are other critical regularities induced by the Galilean invariance for (1.1) and the Lorentzian symmetry for (1.2) below which the equations are ill-posed; see [51,25,56,42]. We point out, however, that these additional critical regularities are relevant only when the dimension is low and/or the degree p is small.…”

“…This critical regularity s crit provides a threshold regularity for well-posedness and ill-posedness of (1.1) and (1.2). 2 While there is no dilation symmetry in the periodic setting, the heuristics provided by the scaling argument also plays an important role. On the one hand, there is a good local-in-time theory for (1.1) and (1.2) (at least when the dimension d and the degree p are not too small).…”