A stochastic control approach to Euclidean field theories with exponential interaction

In this talk, I demonstrate how to obtain couplings of the Liouville field and the sinh-Gordon field with the Gaussian free field in dimension d=2, such that the difference is in a Sobolev space of regularity \alpha>1. The analysis covers the entire L2 phase. The main tool is the variational approach to Euclidean field theories by Barashkov and Gubinelli applied to field theories with exponential interaction. The additional key ingredients are estimates for the short scales of the minimizer of the variational problem and several applications of the Brascamp-Lieb inequality.