On the fi^4_4-problem

In this paper we illustrate the recent techniques developed to treat renormalizable interactions by developing an example which, although implicit in the literature, has not been pointed out. We consider the formal perturbation expansion of $\lambda\varphi^4_4$-hierarchical with $\lambda > 0$ (i.e. the "unstable" case) and show that one can construct a one parameter family $P_\lambda$, $\lambda\in [0,\lambda_0], of real stochastic processes whose effective potential (and Schwinger functions) admit the formal renormalized series as asymptotic series: this shows that $P_\lambda$ can be a family of probability measure and does not have to be necessarily complex valued, as it is in the constructions based on analytic continuation from $\lambda < 0$ to $\lambda > 0$. The construction seems to be generalizable to more interesting hierarchical models (i.e. having non-trivial S-matrix): however as it stands, it is bound to produce in the Euclidean case a theory violating the Osterwalder-Schrader positivity (which would be kept only in the hierarchical cases).