Remarks on non-linear noise excitability of some stochastic heat equations

Abstract

We consider nonlinear parabolic SPDEs of the form ${\partial }_{t}u=\hat{I}”u+\hat{I}»\ddot{I}ƒ(u)\dot{w}$ on the interval $(0,L)$, where $\dot{w}$ denotes space-time white noise, $\ddot{I}ƒ$ is Lipschitz continuous. Under Dirichlet boundary conditions and a linear growth condition on $\ddot{I}ƒ$, we show that the expected $L^2$-energy is of order $\exp [\text{const}\times \hat{I}{»}^4]$ as $\hat{I}»\to \infty$. This significantly improves a recent result of Khoshnevisan and Kim. Our method is very different from theirs and it allows us to arrive at the same conclusion for the same equation but with Neumann boundary condition. This improves over another result of Khoshnevisan and Kim.