A Stochastic Analysis Approach to Tensor Field Theories

We present two different arguments using stochastic analysis to construct super-renormalizable tensor field theories, namely the ${\mathrm{T}}_3^4$ and ${\mathrm{T}}_4^4$ models. The first approach is the construction of a Langevin dynamic combined with a PDE energy estimate while the second is an application of the variational approach of Barashkov and Gubinelli. By leveraging the melonic structure of divergences, regularising properties of non-local products, and controlling certain random operators, we demonstrate that for tensor field theories these arguments can be significantly simplified in comparison to what is required for ${\mathrm{\Phi }}_d^4$ models.