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

Alessandra Faggionato; Davide Gabrielli

Annales de l'I.H.P. Probabilités et statistiques (2012)

- Volume: 48, Issue: 1, page 212-234
- ISSN: 0246-0203

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topFaggionato, Alessandra, and Gabrielli, Davide. "A representation formula for large deviations rate functionals of invariant measures on the one dimensional torus." Annales de l'I.H.P. Probabilités et statistiques 48.1 (2012): 212-234. <http://eudml.org/doc/271970>.

@article{Faggionato2012,

abstract = {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.},

author = {Faggionato, Alessandra, Gabrielli, Davide},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {diffusion; piecewise deterministic Markov process; invariant measure; large deviations; Hamilton–Jacobi equation; Hamilton-Jacobi equation},

language = {eng},

number = {1},

pages = {212-234},

publisher = {Gauthier-Villars},

title = {A representation formula for large deviations rate functionals of invariant measures on the one dimensional torus},

url = {http://eudml.org/doc/271970},

volume = {48},

year = {2012},

}

TY - JOUR

AU - Faggionato, Alessandra

AU - Gabrielli, Davide

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

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2012

PB - Gauthier-Villars

VL - 48

IS - 1

SP - 212

EP - 234

AB - 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.

LA - eng

KW - diffusion; piecewise deterministic Markov process; invariant measure; large deviations; Hamilton–Jacobi equation; Hamilton-Jacobi equation

UR - http://eudml.org/doc/271970

ER -

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