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We consider a generic diffusion on the 1D torus and give a simplerepresentation formula for the large deviation rate functional of its invariantprobability measure, in the limit of vanishing noise. Previously, this ratefunctional had been characterized by M.I. Freidlin and A.D.\ Wentzell assolution of a rather complex optimization problem. We discuss this last problemin full generality and show that it leads to our formula. We express the ratefunctional by means of a geometric transformation that, with a Maxwell-likeconstruction, creates flat regions.  We then consider piecewise deterministic Markov processes on the 1D torus andshow that the corresponding large deviation rate functional for the stationarydistribution is obtained by applying the same transformation. Inspired by this,we prove a universality result showing that the transformation generatesviscosity solution of stationary Hamilton--Jacobi equation associated to anyHamiltonian $H$ satisfying suitable weak conditions.

A representation formula for large deviations rate functionals of invariant measures on the one dimensional torus