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

Abstract

We introduce an explicit description of the ${\Phi }_3^4$ 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, $\Gamma$-convergence allows to extend the variational characterization to the unregularized model.

arXiv comment47 pages, added proof of Gamma-convergence, title changed

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• Quantitative quasi-invariance of Gaussian measures below the energy level for the 1D generalized nonlinear Schrödinger equation and application to global well-posedness
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