  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.
