Non-linear noise excitation and intermittency under high disorder

• Published in: Proceedings of the American Mathematical Society 2015-09, Vol.143 (9), p.4073-4083
• Main Authors: KHOSHNEVISAN, DAVAR, KIM, KUNWOO
• Format: Article
• Language: English
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-2015-12517-8

Consider the semilinear heat equation \partial _t u = \partial ^2_x u + \lambda \sigma (u)\xi on the interval [0\,,L] with Dirichlet zero-boundary condition and a nice non-random initial function, where the forcing \xi is space-time white noise and \lambda >0 denotes the level of the noise. We show that, when the solution is intermittent [that is, when \inf _z\vert\sigma (z)/z\vert>0], the expected L^2-energy of the solution grows at least as \exp \{c\lambda ^2\} and at most as \exp \{c\lambda ^4\} as \lambda \to \infty . In the case that the Dirichlet boundary condition is replaced by a Neumann boundary condition, we prove that the L^2-energy of the solution is in fact of sharp exponential order \exp \{c\lambda ^4\}. We show also that, for a large family of one-dimensional randomly forced wave equations on \mathbf {R}, the energy of the solution grows as \exp \{c\lambda \} as \lambda \to \infty . Thus, we observe the surprising result that the stochastic wave equation is, quite typically, significantly less noise-excitable than its parabolic counterparts.