Non-linear noise excitation and intermittency under high disorder

Davar Khoshnevisan and Kunwoo Kim

Abstract. Consider the semilinear heat equation \(\partial_t u = \partial^2_x u + \lambda\sigma(u)\xi\) on the interval \([0\,,1]\) 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 \(\inf_{z\in{\bf R}}|\sigma(z)/z|>0\)], 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 \(\lambda\to\infty\). When the initial function is bounded uniformly away from zero and the Dirichlet boundary condition is replaced by a Neumann condition, then the \(L^2\)-energy of the solution is in fact shown to be of sharp exponential order \(\exp\{c\lambda^4\}\). We show also that, for a large family of one-dimensional randomly-forced wave equations, 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.

Keywords. The stochastic heat equation; the stochastic wave equation; intermittency; non-linear noise excitation.

AMS Classification (2000). Primary 60H15, 60H25; Secondary 35R60, 60K37, 60J30, 60B15.

Support. Research supported in part by a grant from the U.S. National Science Foundation.

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Davar Khoshnevisan & Kunwoo Kim
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A. &

Last Update: March 4, 2013
© 2013 - Davar Khoshnevisan & Kunwoo Kim