Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: measures

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 $$\Phi ^4_3$$ Φ 3 4 -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.

Stochastic quantization of the $Φ^3_3$-model

“…As in case of the Φ 4 3 -measure in [3], we can prove uniform exponential integrability of the truncated density e −R N (u) in L p (µ) only for p = 1 due to the second renormalization introduced in (1.24). See also [51,12] for a similar phenomenon in the case of the defocusing Hartree Φ 4 3 -measure. We point out that the renormalized potential energy R N (u) in (1.24) does not converge to any limit and neither does the density e −R N (u) , which is essentially the source of the singularity of the Φ 3 3 -measure with respect to the massive Gaussian free field µ.…”

“…For this reason, he describes his new globalization argument as "the probabilistic version of a deterministic global theory using almost conservation laws". We also point out that Bringmann's analysis relies on the fact that the (truncated) Gibbs measure is absolutely continuous with respect to a shifted measure [51,12] (as in Appendix A below).…”

“…As a result, Bourgain's invariant measure argument [8,10] is not directly applicable to our problem. In [13], Bringmann encountered a similar problem in the context of the defocusing Hartree NLW on T 3 , where he overcame this issue by introducing a new globalization argument, by using the fact that the (truncated) Gibbs measure is absolutely continuous with respect to a shifted measure (as in Appendix A below) [51,12] in a uniform manner and establishing a (rather involved) large time stability theory, where sets of large probabilities are characterized via the shifted measures.…”

“…modification of the proof of Theorem 1.14 yields uniform (in N ) local well-posedness of the truncated equation (1.38) (with the same limiting process (u, ∂ t u) as in Theorem 1.14) for the initial data of the form(1.37). By exploiting (formal) invariance of the truncated Gibbs measure ρ N in (1.31),12 we see that the truncated hyperbolic Φ 3 3 -model(1.38) is almost surely globally well-posed with respect to the truncated Gibbs measure ρ N and, moreover, ρ N is invariant under the resulting dynamics; seeLemma 6.4. …”

“…In the same work, we also constructed the invariant Gibbs dynamics for the associated (canonical) stochastic quantization equation. See also [51,12,13] for the defocusing case (σ < 0). Note that when β = 0, the defocusing Hartree Φ 4 3 -measure reduces to the usual Φ 4 3 -measure.…”