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

Abstract

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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.

How to cite

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Faggionato, 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 -

References

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  3. [3] M. Crandall and P. L. Lions. Viscosity solutions of Hamilton–Jacobi equations. Trans. Amer. Math. Soc.277 (1983) 1–42. Zbl0599.35024MR690039
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  7. [7] A. Faggionato, D. Gabrielli and M. Ribezzi Crivellari. Non-equilibrium thermodynamics of piecewise deterministic Markov processes. J. Stat. Phys.137 (2009) 259–204. Zbl1179.82108MR2559431
  8. [8] M. I. Freidlin and A. D. Wentzell. Random Perturbations of Dynamical Systems. Grundlehren der mathematichen Wissenschaften 260. Springer, Berlin, 1984. Zbl0522.60055MR722136
  9. [9] J. R. Gomez-Solano, A. Petrosyan, S. Ciliberto, R. Chetrite and K. Gawedzki. Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state. Phys. Rev. Lett. 103 (2009) 040601. 
  10. [10] Y. Kifer. Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging. Mem. Amer. Math. Soc.201 (2009) 1–129. Zbl1222.37002MR2547839
  11. [11] C. Maes, K. Netočný and B. Wynants. Steady state statistics of driven diffusions. Phys. A387 (2008) 2675–2689. MR2587167

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