  In this outline of a program, based on rigorous renormalization group theory,
we introduce new definitions which allow one to formulate precise mathematical
conjectures related to conformal invariance as studied by physicists in the
area known as higher-dimensional conformal bootstrap which has developed at a
breathtaking pace over the last few years. We also explore a second theme,
intimately tied to conformal invariance for random distributions, which can be
construed as a search for very general first and second-quantized
Kolmogorov-Chentsov Theorems. First-quantized refers to regularity of sample
paths. Second-quantized refers to regularity of generalized functionals or Hida
distributions and relates to the operator product expansion. We formulate this
program in both the Archimedean and $p$-adic situations. Indeed, the study of
conformal field theory and its connections with probability provides a golden
opportunity where $p$-adic analysis can lead the way towards a better
understanding of open problems in the Archimedean setting. Finally, we present
a summary of progress made on a $p$-adic hierarchical model and point out
possible connections to number theory. Parts of this article were presented in
author's talk at the 6th International Conference on $p$-adic Mathematical
Physics and its Applications, Mexico 2017.
