Abstract
In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The main novelty is the singularity of the Gibbs measure with respect to the Gaussian free field. The singularity has several consequences in both measure-theoretic and dynamical aspects of our argument. In this paper, we construct and study the Gibbs measure. Our approach is based on earlier work of Barashkov and Gubinelli for the Phi ^4_3-model. Most importantly, our truncated Gibbs measures are tailored towards the dynamical aspects in the second part of the series. In addition, we develop new tools dealing with the non-locality of the Hartree interaction. We also determine the exact threshold between singularity and absolute continuity of the Gibbs measure depending on the regularity of the interaction potential.
Highlights
Introduction to the seriesIn this two-paper series, we study the renormalized wave equation with a Hartree nonlinearity and random initial data given by−∂t2t u − u + u = :(V ∗ u2)u : (t, x) ∈ R × T3,(a) u|t=0 = φ0, ∂t u|t=0 = φ1.Here, T d=ef R/2π Z is the torus and the interaction potential V : T3 → R is of the form V (x) = cβ |x|−(3−β) for all small x ∈ T3, where 0 < β < 3, satisfies V (x) 1 for all x ∈ T3, is even, and is smooth away from the origin
The renormalized wave equation with Hartree nonlinearity (a) is globally well-posed on the support of μ⊗ and the dynamics leave μ⊗ invariant. This is the first example of an invariant Gibbs measure for a dispersive equation which is singular with respect to the Gaussian free field g⊗
2 L2 dt Recalling the definition of ltT from Proposition 3.1 and (4.26), we obtain that ltT = htT,w + λ JtT :(V ∗ (WtT,uT )2)we have that (WtT),uT :
Summary
In this two-paper series, we study the renormalized wave equation with a Hartree nonlinearity and random initial data given by. We rely on ideas from the para-controlled calculus of Gubinelli, Imkeller, and Perkowski [20] The heart of this series, lies in the global theory. The formal Gibbs measure μ⊗ exists and, for 0 < β < 1/2, is singular with respect to the Gaussian free field g⊗. The renormalized wave equation with Hartree nonlinearity (a) is globally well-posed on the support of μ⊗ and the dynamics leave μ⊗ invariant. This is the first example of an invariant Gibbs measure for a dispersive equation which is singular with respect to the Gaussian free field g⊗
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