Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Large deviations principles for stochastic scalar conservation laws
Download PDF
Download PDF
  • Published: 23 April 2009

Large deviations principles for stochastic scalar conservation laws

  • Mauro Mariani1 

Probability Theory and Related Fields volume 147, pages 607–648 (2010)Cite this article

  • 326 Accesses

  • 31 Citations

  • Metrics details

Abstract

Large deviations principles for a family of scalar 1 + 1 dimensional conservative stochastic PDEs (viscous conservation laws) are investigated, in the limit of jointly vanishing noise and viscosity. A first large deviations principle is obtained in a space of Young measures. The associated rate functional vanishes on a wide set, the so-called set of measure-valued solutions to the limiting conservation law. A second order large deviations principle is therefore investigated, however, this can be only partially proved. The second order rate functional provides a generalization for non-convex fluxes of the functional introduced by Jensen and Varadhan in a stochastic particles system setting.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Ambrosio, L., De Lellis, C., Maly, J.: On the chain rule for the divergence of BV like vector fields: applications, partial results, open problems. AMS series in contemporary mathematics “Perspectives in Nonlinear Partial Differential Equations: in honor of Haim Brezis” (2005)

  2. Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2000)

    MATH  Google Scholar 

  3. Bellettini, G., Bertini, L., Mariani, M., Novaga, N.: Γ-entropy cost functional for scalar conservation laws (to appear in Arch.Rat.Mech.Anal.)

  4. Billingsley P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)

    MATH  Google Scholar 

  5. Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics, 2nd edn. Springer, Berlin (2005)

    MATH  Google Scholar 

  6. De Lellis C., Otto F., Westdickenberg M.: Structure of entropy solutions for multi-dimensional scalar conservation laws. Arch. Ration. Mech. Anal. 170(2), 137–184 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  8. Dembo A., Zeitouni O.: Large Deviations Techniques and Application. Springer, New York (1993)

    Google Scholar 

  9. Feng, J., Kurtz, T.G.: Large deviations for stochastic processes. Mathematical Surveys and Monographs, vol. 131, American Mathematical Society (2006)

  10. Freidlin M.I., Wentzell A.D.: Random Perturbations of Dynamical Systems, 2nd edn. Springer, New York (1998)

    MATH  Google Scholar 

  11. Jakubowski A.: On the Skorokhod topology. Ann. Inst. H. Poincar Probab. Statist. 22(3), 263–285 (1986)

    MATH  MathSciNet  Google Scholar 

  12. Jensen, L.H.: Large deviations of the asymmetric simple exclusion process in one dimension. Ph.D. Thesis, Courant Institute NYU (2000)

  13. Kipnis C., Landim C.: Scaling Limits of Interacting Particle Systems. Springer, Berlin (1999)

    MATH  Google Scholar 

  14. Karatzas I., Shreve S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1988)

    MATH  Google Scholar 

  15. Landim C.: Hydrodynamical limit for space inhomogeneous one dimensional totally asymmetric zero range process. Ann. Probab. 24(2), 599–638 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lions P.-L., Souganidis P.: Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Sr. I Math. 326(9), 1085–1092 (1998)

    MATH  MathSciNet  Google Scholar 

  17. Lions P.-L., Souganidis P.: Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Sr. I Math. 331(10), 783–790 (2000)

    MATH  MathSciNet  Google Scholar 

  18. Mariani, M.: Large Deviations for stochastic conservation laws and their variational counterparts, Ph.D. Thesis, Sapienza Università di Roma (2007)

  19. Revuz D., Yor M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1999)

    MATH  Google Scholar 

  20. Spohn H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)

    MATH  Google Scholar 

  21. Varadhan S.R.S.: Large deviations for the simple asymmetric exclusion process. Stochastic analysis on large scale interacting systems. Adv. Stud. Pure Math. 39, 1–27 (2004)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CEREMADE, UMR-CNRS 7534, Université de Paris-Dauphine, Place du Marechal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Mauro Mariani

Authors
  1. Mauro Mariani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mauro Mariani.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mariani, M. Large deviations principles for stochastic scalar conservation laws. Probab. Theory Relat. Fields 147, 607–648 (2010). https://doi.org/10.1007/s00440-009-0218-6

Download citation

  • Received: 01 July 2008

  • Revised: 11 March 2009

  • Published: 23 April 2009

  • Issue Date: July 2010

  • DOI: https://doi.org/10.1007/s00440-009-0218-6

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stochastic PDE
  • Large deviations
  • Conservation laws
  • Entropy functional

Mathematics Subject Classification (2000)

  • 60H15
  • 60F10
  • 60K35
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature