Skip to main content
SpringerLink
Log in

Non-local Diffusions, Drifts and Games

  • Conference paper
Book cover Nonlinear Partial Differential Equations

Part of the book series: Abel Symposia ((ABEL,volume 7))

Abstract

This is a brief discussion of the properties of solutions to several non-linear elliptic equations involving diffusive processes of non-local nature. These equation arise in several contexts: from continuum mechanics and phase transition, from population dynamics, from optimal control and game theory. The equations coming from continuum mechanics exhibit a variational structure and a theory parallel to the De Giorgi–Nash–Moser was necessary to show existence of regular solutions. Population dynamics suggests “porous media like equations” with a non-local pressure, and from optimal control we obtain fully non-linear equations that require methods of the type of the Krylov–Safonov–Evans theory. Finally, we discuss some non-local p and infinite Laplacian models coming from game theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Barlow, M.T., Bass, R.F., Chen, Zh.Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. 361(4), 1963–1999 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bjorland, C., Caffarelli, L., Figalli, A.: Non-local tug-of-war and the infinity fractional Laplacian. arXiv:1011.1966

  3. Bass, R.F., Kassmann, M.: Harnack inequalities for non-local operators of variable order. Trans. Am. Math. Soc. 357(2), 837–850 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Caffarelli, L., Cabre, X.: Fully Nonlinear Elliptic Equations. Am. Math. Soc. Colloq. Publ., vol. 43. Am. Math. Soc., Providence (1995)

    MATH  Google Scholar 

  5. Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (2011, in press)

    Google Scholar 

  6. Caffarelli, L.A., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caffarelli, L.A., Silvestre, L.: Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal. (2011, in press). arXiv:0902.4030

  8. Caffarelli, L.A., Silvestre, L.: The Evans-Krylov theorem for non local fully non linear equations. Ann. Math. (2011, in press). arXiv:0905.1339

  9. Caffarelli, L.A., Chan, C.H., Vasseur, A.: Regularity theory for nonlinear integral operators. arXiv:1003.1699

  10. Chambolle, A., Lindgren, E., Monneau, R.: The Hölder infinite Laplacian and Hölder extensions. Preprint (2010)

    Google Scholar 

  11. Duvant, G., Lions, J.-L.: Les inéquations en mécanique et en physique. Travaux et Recherches Mathématiques, vol. 21. Dunod, Paris (1972) (in French)

    Google Scholar 

  12. Giacomin, G., Lebowitz, J.L., Presutti, E.: Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems. In: Stochastic Partial Differential Equations: Six Perspectives. Math. Surveys Monogr., vol. 64, pp. 107–152. Am. Math. Soc., Providence (1999)

    Google Scholar 

  13. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kassmann, M.: A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial Differ. Equ. 34(1), 1–21 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Komatsu, T.: Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math. 32(4), 833–860 (1995)

    MathSciNet  MATH  Google Scholar 

  16. Peres, Y., Sheffield, S.: Tug-of-war with noise: a game-theoretic view of the p-Laplacian. Duke Math. J. 145(1), 91–120 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B.: Tug-of-war and the infinity Laplacian. J. Am. Math. Soc. 22(1), 167–210 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author was partially supported by National Science Foundation Grant DMS-0654267.

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, 1 University Station—C1200, Austin, TX, 78712, USA

    Luis Caffarelli

Authors
  1. Luis Caffarelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luis Caffarelli .

Editor information

Editors and Affiliations

  1. of Science and Technology, Dept. of Mathematics, Norwegian University, Trondheim, 7491, Norway

    Helge Holden

  2. Centre of Mathematics for Applications, University of Oslo, Oslo, 0316, Norway

    Kenneth H. Karlsen

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Caffarelli, L. (2012). Non-local Diffusions, Drifts and Games. In: Holden, H., Karlsen, K. (eds) Nonlinear Partial Differential Equations. Abel Symposia, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25361-4_3

Download citation

Publish with us

Policies and ethics

Search

Navigation