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

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 solu...

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Bibliographic Details
Published inAnnales de l'I.H.P. Probabilités et statistiques Vol. 48; no. 1; pp. 212 - 234
Main Authors FAGGIONATO, Alessandra, GABRIELLI, Davide
Format Journal Article
LanguageEnglish
Published Paris Elsevier 01.02.2012
Institut Henri Poincaré
Subjects
60F10
60J60
82C05
Diffusion
Exact sciences and technology
General topics
Hamilton–Jacobi equation
Invariant measure
Large deviations
Limit theorems
Markov processes
Mathematics
Piecewise deterministic Markov process
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Statistics
Markov process
82C05
60F10
60J60
Probability theory
Rate function
Hamilton―Jacobi equation
Large deviations
Optimization
Large deviation
Functional
Hamilton Jacobi equation
Piecewise deterministic Markov process
Invariant measure
Viscosity solution
Diffusion
Hamiltonian
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Summary: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.
ISSN:0246-0203
DOI:10.1214/10-AIHP412