A variational method for ${\mathrm{\Phi }}_3^4$

We introduce an explicit description of the ${\mathrm{\Phi }}_3^4$ measure on a bounded domain. Our starting point is the interpretation of its Laplace transform as the value function of a stochastic optimal control problem along the flow of a scale regularization parameter. Once small scale singularities have been renormalized by the standard counterterms, $\mathrm{\Gamma }$-convergence allows to extend the variational characterization to the unregularized model.