On the valleys of the stochastic heat equation

Abstract

We consider a generalization of the parabolic Anderson model driven by space-time white noise, also called the stochastic heat equation, on the real line:

$${\partial }_{\mathit{t}}\mathit{u}(\mathit{t},\mathit{x})=\frac{1}{2}{\partial }_{\mathit{x}}^2\mathit{u}(\mathit{t},\mathit{x})\mathbf{+}\mathit{\sigma }(\mathit{u}(\mathit{t},\mathit{x}))\mathit{\xi }(\mathit{t},\mathit{x})\text{for}\mathit{t}>0\text{and}\mathit{x}\in \mathbb{R}.$$

High peaks of solutions have been extensively studied under the name of intermittency, but less is known about spatial regions between peaks, which we may loosely refer to as valleys. We present two results about the valleys of the solution.

Our first theorem provides information about the size of valleys and the supremum of the solution $\mathit{u}(\mathit{t},\mathit{x})$ over a valley. More precisely, when the initial function ${\mathit{u}}_0(\mathit{x})=1$ for all $\mathit{x}\in \mathbb{R}$, we show that the supremum of the solution over a valley vanishes as $\mathit{t}\to \infty$, and we establish an upper bound of $exp\{-\text{const}·{\mathit{t}}^{1/3}\}$ for $\mathit{u}(\mathit{t},\mathit{x})$ when x lies in a valley. We demonstrate also that the length of a valley grows at least as $exp\{\mathbf{+}\text{const}·{\mathit{t}}^{1/3}\}$ as $\mathit{t}\to \infty$.

Our second theorem asserts that the length of the valleys are eventually infinite when the initial function $\mathit{u}(0,\mathit{x})$ has subgaussian tails.