We introduce an explicit description of the $\Phi^4_3$ 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, $\Gamma$-convergence allows to extend the variational characterization to the unregularized model.
Categories: math.PR math-ph math.MP