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In this article we provide the complete proof of the result announced inarXiv:1210.7717 about the construction of scale invariant non-Gaussiangeneralized stochastic processes over three dimensional p-adic space. Theconstruction includes that of the associated squared field and our result showsthis squared field has a dynamically generated anomalous dimension whichrigorously confirms a prediction made more than forty years ago, in anessentially identical situation, by K. G. Wilson. We also prove a mild form ofuniversality for the model under consideration. Our main innovation is that ourrigourous renormalization group formalism allows for space dependent couplings.We derive the relationship between mixed correlations and the dynamical systemsfeatures of our extended renormalization group transformation at a nontrivialfixed point. The key to our control of the composite field is a partiallinearization theorem which is an infinite-dimensional version of the KoenigsTheorem in holomorphic dynamics. This is akin to a nonperturbative constructionof a nonlinear scaling field in the sense of F. J. Wegner infinitesimally nearthe critical surface. Our presentation is essentially self-contained and gearedtowards a wider audience. While primarily concerning the areas of probabilityand mathematical physics we believe this article will be of interest toresearchers in dynamical systems theory, harmonic analysis and number theory.It can also be profitably read by graduate students in theoretical physics witha craving for mathematical precision while struggling to learn therenormalization group.

Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions