Intermittency and Chaos for a Nonlinear Stochastic Wave Equation in Dimension 1
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).
KeywordsIntermittency 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.
- 2.Assing, S.: A rigorous equation for the Cole–Hopf solution of the conservative KPZ dynamics. Preprint (2011) available from http://arxiv.org/abs/1109.2886
- 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
- 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
- 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
- 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
- 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
- 22.Gonçalves, P., Jara, M.: Universality of KPZ equation. Preprint (2010) available from http://arxiv.org/abs/1003.4478
- 23.Hairer, M.: Solving the KPZ equation. To appear in Annals of Mathematics. (2012) Available from http://arxiv.org/abs/1109.6811
- 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