Non-linear noise excitation and intermittency under high disorder

TL;DR

This paper investigates how solutions to certain stochastic partial differential equations respond to high levels of noise, revealing that parabolic equations exhibit exponential energy growth with noise level, while wave equations are less sensitive.

Contribution

It provides new bounds on the energy growth of solutions to semilinear heat and wave equations under high noise levels, highlighting differences between parabolic and hyperbolic cases.

Findings

Energy grows at least as exp(cλ^2) and at most as exp(cλ^4) for the heat equation with Dirichlet boundary.

Energy growth for Neumann boundary conditions is sharply exp(cλ^4).

Wave equations show energy growth as exp(cλ), indicating less noise sensitivity.

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 $\inf_z|\sigma(z)/z|>0$], 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 $\lambda\to\infty$. 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, 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.