A coupling for the Liouville and the sinh-Gordon model in the $L^2$ phase

Using a stochastic control approach we establish 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 $L^2$ phase. Our main tools are estimates for the short scales of the minimiser of the variational problem and several applications of the Brascamp-Lieb inequality.