Intermittency and Chaos for a Nonlinear Stochastic Wave Equation in Dimension 1

  • Daniel ConusEmail author
  • Mathew Joseph
  • Davar Khoshnevisan
  • Shang-Yuan Shiu
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)


Consider a nonlinear stochastic wave equation driven by space-time white noise in dimension one. We discuss the intermittency of the solution, and then use those intermittency results in order to demonstrate that in many cases the solution is chaotic. For the most part, the novel portion of our work is about the two cases where (1) the initial conditions have compact support, where the global maximum of the solution remains bounded, and (2) the initial conditions are positive constants, where the global maximum is almost surely infinite. Bounds are also provided on the behavior of the global maximum of the solution in Case (2).


Intermittency The stochastic wave equation Chaos 



An anonymous referee read this paper quite carefully and made a number of critical suggestions and corrections that have improved the paper. We thank him or her wholeheartedly.


  1. 1.
    Amir G., Corwin I., Quastel J.: Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Comm. Pure Appl. Math. 64, 466–537 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Assing, S.: A rigorous equation for the Cole–Hopf solution of the conservative KPZ dynamics. Preprint (2011) available from
  3. 3.
    Bertini, L., Cancrini N.: The stochastic heat equation: Feynman-Kac formula and intermittence. J. Stat. Phys. 78(5–6), 1377–1401 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Burkholder, D.L.: Martingale transforms. Ann. Math. Stat. 37, 1494–1504 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Burkholder, D.L., Gundy, R.F.: Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124, 249–304 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Burkholder, D.L., Davis B.J., Gundy, R.F.: Integral inequalities for convex functions of operators on martingales. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. II, pp. 223–240. University of California Press, Berkeley, California (1972)Google Scholar
  7. 7.
    Carlen, E., Kree, P.: L p estimates for multiple stochastic integrals. Ann. Probab. 19(1), 354–368 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Carmona, R.A., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Memoires of the American Mathematical Society, vol. 108. American Mathematical Society, Rhode Island (1994)Google Scholar
  9. 9.
    Carmona, R.A., Nualart, D.: Random nonlinear wave equations: propagation of singularities. Ann. Probab. 16(2), 730–751 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Conus, D., Dalang, R.C.: The non-linear stochastic wave equation in high dimensions. Electron. J. Probab. 13, 629–670 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Conus, D., Khoshnevisan, D.: On the existence and position of the farthest peaks of a family of stochastic heat and wave equations. Probab. Theory Relat. Fields. 152 n.3–4, 681–701 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Conus, D., Joseph, M., Khoshnevisan, D.: On the chaotic character of the stochastic heat equation, before the onset of intermittency. Ann. Probab. To appear.Google Scholar
  13. 13.
    Dalang, R.C.: Extending martingale measure stochastic integral with applications to spatially homogeneous spde’s. Electron. J. Probab. 4, 1–29 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dalang, R.C., Frangos, N.E.: The stochastic wave equation in two spatial dimensions. Ann. Probab. 26(1), 187–212 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dalang, R.C., Mueller, C.: Some non-linear s.p.d.e’s that are second order in time. Electron. J. Probab. 8 (2003)Google Scholar
  16. 16.
    Dalang, R.C., Mueller, C.: Intermittency properties in a hyperbolic Anderson problem. Ann. l’Institut Henri Poincaré 45(4), 1150–1164 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Dalang, R.C., Khohsnevisan, D., Mueller, C., Nualart, D., Xiao, Y.: A minicourse in stochastic partial differential equations. In: Lecture Notes in Mathematics, vol. 1962. Springer, Berlin (2006)Google Scholar
  18. 18.
    Davis, B.: On the L p norms of stochastic integrals and other martingales. Duke Math. J. 43(4), 697–704 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Foondun, M., Khoshnevisan, D.: Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14(12), 548–568 (2009)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Foondun, M., Khoshnevisan, D.: On the global maximum of the solution to a stochastic heat equation with compact-support initial data. Ann. l’Institut Henri Poincaré 46(4), 895–907 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gärtner, J., König, W., Molchanov S.: Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35(2), 439–499 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Gonçalves, P., Jara, M.: Universality of KPZ equation. Preprint (2010) available from
  23. 23.
    Hairer, M.: Solving the KPZ equation. To appear in Annals of Mathematics. (2012) Available from
  24. 24.
    Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1985)CrossRefGoogle Scholar
  25. 25.
    Mueller, C.: On the support of solutions to the heat equation with noise. Stochast. Stochast. Rep. 37(4), 225–245 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Walsh, J.B.: An introduction to stochastic partial differential equations. In: Ecole d’Etè de Probabilitès de St-Flour, XIV, 1984. Lecture Notes in Mathematics, vol. 1180, pp. 265–439. Springer, Berlin (1986)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Conus
    • 1
    Email author
  • Mathew Joseph
    • 2
    • 3
  • Davar Khoshnevisan
    • 2
  • Shang-Yuan Shiu
    • 4
  1. 1.Department of MathematicsLehigh UniversityBethlehemUSA
  2. 2.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  3. 3.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK
  4. 4.Department of MathematicsNational Central UniversityJhongli CityTaiwan

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