Abstract
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).
Keywords
- Intermittency
- The stochastic wave equation
- Chaos
MSC Subject Classication 2000: 60H15; 60H20, 60H05
Received 12/5/2011; Accepted 2/29/2012; Final 7/17/2012
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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.
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This paper is dedicated to Professor David Nualart, whose scientific innovations have influenced us greatly.
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Conus, D., Joseph, M., Khoshnevisan, D., Shiu, SY. (2013). Intermittency and Chaos for a Nonlinear Stochastic Wave Equation in Dimension 1. In: Viens, F., Feng, J., Hu, Y., Nualart , E. (eds) Malliavin Calculus and Stochastic Analysis. Springer Proceedings in Mathematics & Statistics, vol 34. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5906-4_11
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