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Abstract
Lebowitz, Rose, and Speer (1988) initiated the study of focusing Gibbs measures, which was continued by Brydges and Slade (1996), Bourgain (1997, 1999), and Carlen,Fröhlich, and Lebowitz (2016) among others. In this paper, we complete the program on the (non)construction of the focusing Hartree Gibbs measures in the threedimensional setting. More precisely, we study a focusing Φ4/3model with a Hartreetype nonlinearity, where the potential for the Hartree nonlinearity is given by the Bessel potential of order β. We first construct the focusing Hartree Φ4/3measure for β > 2, while we prove its nonnormalizability for β < 2. Furthermore, we establish the following phase transition at the critical value β = 2: normalizability in the weakly nonlinear regime and nonnormalizability in the strongly nonlinear regime. We then study the canonical stochastic quantization of the focusing Hartree Φ4/3measure, namely, the threedimensional stochastic damped nonlinear wave equation (SdNLW) with a cubic nonlinearity of Hartreetype, forced by an additive spacetime white noise, and prove almost sure global wellposedness and invariance of the
focusing Hartree Φ4/3measure for β > 2 (and β = 2 in the weakly nonlinear regime). In view of the nonnormalizability result, our almost sure global wellposedness result is sharp. In Appendix, we also discuss the (parabolic) stochastic quantization for the focusing Hartree Φ4/3measure.
We also consider the defocusing case. By adapting our argument from the focusing case, we first construct the defocusing Hartree Φ4/3measure and the associated invariant dynamics for the defocusing Hartree SdNLW for β > 1. By introducing further renormalizations at β = 1 and β = ½, we extend the construction of the defocusing Hartree Φ4/3measure for β > 0,
where the resulting measure is shown to be singular with respect to the reference Gaussian free field for 0 < β ≤½.
focusing Hartree Φ4/3measure for β > 2 (and β = 2 in the weakly nonlinear regime). In view of the nonnormalizability result, our almost sure global wellposedness result is sharp. In Appendix, we also discuss the (parabolic) stochastic quantization for the focusing Hartree Φ4/3measure.
We also consider the defocusing case. By adapting our argument from the focusing case, we first construct the defocusing Hartree Φ4/3measure and the associated invariant dynamics for the defocusing Hartree SdNLW for β > 1. By introducing further renormalizations at β = 1 and β = ½, we extend the construction of the defocusing Hartree Φ4/3measure for β > 0,
where the resulting measure is shown to be singular with respect to the reference Gaussian free field for 0 < β ≤½.
Original language  English 

Number of pages  126 
Journal  Memoirs of the american mathematical society 
Publication status  Accepted/In press  13 Jun 2022 
Keywords
 Hartree \Phi^4_3measure
 stochastic quantization
 stochastic nonlinear wave equation
 nonlinear wave equation
 Gibbs measure
 paracontrolled calculus
 nonlinear heat equation
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ProbDynDispEq  Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
1/03/15 → 29/02/20
Project: Research