Construction of a non-Gaussian and rotation-invariant $\Phi ^4$-measure and associated flow on ${\mathbb R}^3$ through stochastic quantization

Abstract

A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures $\mu$ associated with the $\varphi ^4_3$-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 $\varphi ^4_3$-model. Our starting point is a suitable approximation $\mu_{M,N}$ of the measure $\mu$ we intend to construct. $\mu_{M,N}$ is parametrized by an $M$-dependent space cut-off function $\rho_M: {\mathbb R}^3\rightarrow {\mathbb R}$ and an $N$-dependent momentum cut-off function $\psi_N: \widehat{\mathbb R}^3 \cong {\mathbb R}^3 \rightarrow {\mathbb R}$, that act on the interaction term (nonlinear term and counterterms). The corresponding family of stochastic quantization equations yields solutions $(X_t^{M,N}, t\geq 0)$ that have $\mu_{M,N}$ 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 $(X_t^{M,N},t \geq 0)_{M,N}$. Limit points in the sense of convergence in law exist, when both $M$ and $N$ diverge to $+\infty$. The limit processes $(X_t; t\geq 0)$ are continuous on the intersection of suitable Besov spaces and any limit point $\mu$ of the $\mu_{M,N}$ is a stationary measure of $X$. $\mu$ is shown to be a rotation-invariant and non-Gaussian probability measure and we provide results on its support. It is also proven that $\mu$ satisfies a further important property belonging to the family of axioms for Euclidean quantum fields, it is namely reflection positive.