Focusing Φ⁴₃-model with a Hartree-type Nonlinearity

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 three-dimensional setting. More precisely, we study a focusing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi 3 Superscript 4"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Phi ^4_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-model with a Hartree-type nonlinearity, where the potential for the Hartree nonlinearity is given by the Bessel potential of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi>β</mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We first construct the focusing Hartree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi 3 Superscript 4"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Phi ^4_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-measure for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than 2"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta &gt; 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, while we prove its non-normalizability for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than 2"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta &gt; 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Furthermore, we establish the following phase transition at the critical value <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta equals 2"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. We then study the canonical stochastic quantization of the focusing Hartree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi 3 Superscript 4"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Phi ^4_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-measure, namely, the three-dimensional stochastic damped nonlinear wave equation (SdNLW) with a cubic nonlinearity of Hartree-type, forced by an additive space-time white noise, and prove almost sure global well-posedness and invariance of the focusing Hartree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi 3 Superscript 4"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Phi ^4_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-measure for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than 2"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta &gt; 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta equals 2"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the weakly nonlinear regime). In view of the non-normalizability result, our almost sure global well-posedness result is sharp. In Appendix, we also discuss the (parabolic) stochastic quantization for the focusing Hartree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi 3 Superscript 4"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Phi ^4_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-measure. We also consider the defocusing case. By adapting our argument from the focusing case, we first construct the defocusing Hartree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi 3 Superscript 4"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Phi ^4_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-measure and the associated invariant dynamics for the defocusing Hartree SdNLW for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta &gt; 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By introducing further renormalizations at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta equals 1"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta equals one half"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta = \frac 12</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we extend the construction of the defocusing Hartree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi 3 Superscript 4"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Phi ^4_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-measure for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta &gt; 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where the resulting measure is shown to be singular with respect to the reference Gaussian free field for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than beta less-than-or-equal-to one half"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>β</mml:mi> <mml:mo>≤</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">0 &gt; \beta \le \frac 12</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.