Advertisement

The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps

  • Franziska Flegel
  • Martin HeidaEmail author
Article
  • 26 Downloads

Abstract

We study a general class of discrete p-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional p-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Laplace operator to the continuous fractional Laplace operator.

Mathematics Subject Classification

80M40 60H25 60K37 35B27 35R60 47B80 47A75 

Notes

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Bella, P., Fehrman, B., Fischer, J., Otto, F.: Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors. SIAM J. Math. Anal. 49(6), 4658–4703 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Biskup, M.: Recent progress on the random conductance model. Probab. Surv. 8, 294–373 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Boivin, D., Depauw, J.: Spectral homogenization of reversible random walks on \(\mathbb{Z}^d\) in a random environment. Stoch. Process. Appl. 104(1), 29–56 (2003)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bouchaud, J.-P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195(4–5), 127–293 (1990)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, X., Kumagai, T., Wang, J.: Random conductance models with stable-like jumps I: quenched invariance principle. (2018). arXiv:1805.04344
  7. 7.
    Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Progress in Nonlinear Differential Equations and Their Applications, vol. 8. Birkhäuser Boston, Inc., Boston (1993)zbMATHGoogle Scholar
  8. 8.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer, New York (1988)zbMATHGoogle Scholar
  9. 9.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Flegel, F., Heida, M., Slowik, M.: Homogenization theory for a class of random walks among degenerate ergodic weights with long-range jumps. Accepted by AIHP B (2018). 55(3), 1226–1257 (2019)zbMATHGoogle Scholar
  11. 11.
    Heida, M.: On convergences of the squareroot approximation scheme to the Fokker–Planck operator. Math. Models Methods Appl. Sci. 28(13), 2599–2635 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Iannizzotto, A., Liu, S., Perera, K., Squassina, M.: Existence results for fractional p-Laplacian problems via Morse theory. Adv. Calc. Var. 9(2), 101–125 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Jikov, V.V., Kozlov, S.M., Oleĭnik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994). Translated from the Russian by G. A. Yosifian [G. A. Iosif\(^{\prime }\)yan]CrossRefGoogle Scholar
  14. 14.
    Kassmann, M., Piatnitski, A., Zhizhina, E.: Homogenization of lévy-type operators with oscillating coefficients. (2018). arXiv:1807.04371
  15. 15.
    Kozlov, S.M.: Averaging of difference schemes. Math. USSR Sb. 57(2), 351 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Künnemann, R.: The diffusion limit for reversible jump processes onZd with ergodic random bond conductivities. Commun. Math. Phys. 90(1), 27–68 (1983)CrossRefGoogle Scholar
  17. 17.
    Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20(1), 7–51 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Neukamm, S., Schäffner, M., Schlömerkemper, A.: Stochastic homogenization of nonconvex discrete energies with degenerate growth. SIAM J. Math. Anal. 49(3), 1761–1809 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. Random Fields, Vol. I, II (Esztergom, 1979). Colloquia Mathematica Societatis János Bolyai, vol. 27, pp. 835–873. North-Holland, Amsterdam (1981)Google Scholar
  20. 20.
    Piatnitski, A., Zhizhina, E.: Periodic homogenization of nonlocal operators with a convolution-type kernel. SIAM J. Math. Anal. 49(1), 64–81 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Piatnitski, A., Zhizhina, E.: Stochastic homogenization of convolution type operators. (2018). arXiv:1806.00995
  22. 22.
    Tempel’man, A.A.: Ergodic theorems for general dynamical systems. Tr. Mosk. Mat. Obs. 26, 95–132 (1972)MathSciNetGoogle Scholar
  23. 23.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, second edn. Johann Ambrosius Barth, Heidelberg (1995)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  2. 2.Technical University MunichGarchingGermany

Personalised recommendations