Mathematics – Probability
Scientific paper
2010-01-28
Mathematics
Probability
25 pages, 5 figures. Modified title and abstract. Simplified proofs. Additional explanatory figures. Some extentions
Scientific paper
We consider a generic diffusion on the 1D torus and give a simple representation formula for the large deviation rate functional of its invariant probability measure, in the limit of vanishing noise. Previously, this rate functional had been characterized by M.I. Freidlin and A.D.\ Wentzell as solution of a rather complex optimization problem. We discuss this last problem in full generality and show that it leads to our formula. We express the rate functional by means of a geometric transformation that, with a Maxwell-like construction, creates flat regions. We then consider piecewise deterministic Markov processes on the 1D torus and show that the corresponding large deviation rate functional for the stationary distribution is obtained by applying the same transformation. Inspired by this, we prove a universality result showing that the transformation generates viscosity solution of stationary Hamilton--Jacobi equation associated to any Hamiltonian $H$ satisfying suitable weak conditions.
Faggionato Alessandra
Gabrielli Davide
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