Construction of a Non-Gaussian and Rotation-Invariant Φ⁴-Measure and Associated Flow on ℝ³ Through Stochastic Quantization

A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi 3 Superscript 4"> <mml:semantics> <mml:msubsup> <mml:mi>φ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\varphi ^4_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-model of quantum field theory is presented. Our construction uses a combination of semigroup methods, and methods of stochastic partial differential equations (SPDEs) for finding solutions and stationary measures of the natural stochastic quantization associated with the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi 3 Superscript 4"> <mml:semantics> <mml:msubsup> <mml:mi>φ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\varphi ^4_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-model. Our starting point is a suitable approximation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript upper M comma upper N"> <mml:semantics> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mu _{M,N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which we intend to construct. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript upper M comma upper N"> <mml:semantics> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mu _{M,N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is parametrized by an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dependent space cut-off function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript upper M Baseline colon double-struck upper R cubed right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>M</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\rho _M: {\mathbb R}^3\rightarrow {\mathbb R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dependent momentum cut-off function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi Subscript upper N Baseline colon ModifyingAbove double-struck upper R With caret cubed approximately-equals double-struck upper R cubed right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ψ</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>≅</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\psi _N: \widehat {\mathbb R}^3 \cong {\mathbb R}^3 \rightarrow {\mathbb R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that act on the interaction term (nonlinear term and counterterms). The corresponding family of stochastic quantization equations yields solutions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X Subscript t Superscript upper M comma upper N Baseline comma t greater-than-or-equal-to 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X_t^{M,N}, t\geq 0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that have <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript upper M comma upper N"> <mml:semantics> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mu _{M,N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as an invariant probability measure. By a combination of probabilistic and functional analytic methods for singular stochastic differential equations on negative-indices weighted Besov spaces (with rotation invariant weights) we prove the tightness of the family of continuous processes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X Subscript t Superscript upper M comma upper N Baseline comma t greater-than-or-equal-to 0 right-parenthesis Subscript upper M comma upper N"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(X_t^{M,N},t \geq 0)_{M,N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Limit points in the sense of convergence in law exist, when both <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</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="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> diverge to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="plus normal infinity"> <mml:semantics> <mml:mrow> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">+\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The limit processes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X Subscript t Baseline semic