February 2024 On the valleys of the stochastic heat equation
Davar Khoshnevisan, Kunwoo Kim, Carl Mueller
Ann. Appl. Probab. 34(1B): 1177-1198 (February 2024). DOI: 10.1214/23-AAP1988

## 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:

${\partial }_{\mathit{t}}\mathit{u}\left(\mathit{t},\mathit{x}\right)=\frac{1}{2}{\partial }_{\mathit{x}}^{2}\mathit{u}\left(\mathit{t},\mathit{x}\right)\mathbf{+}\mathit{\sigma }\left(\mathit{u}\left(\mathit{t},\mathit{x}\right)\right)\mathit{\xi }\left(\mathit{t},\mathit{x}\right)\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{2.5pt}{0ex}}\mathit{t}>0\phantom{\rule{2.5pt}{0ex}}\text{and}\phantom{\rule{2.5pt}{0ex}}\mathit{x}\in \mathbb{R}.$

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 $\mathit{u}\left(\mathit{t},\mathit{x}\right)$ over a valley. More precisely, when the initial function ${\mathit{u}}_{0}\left(\mathit{x}\right)=1$ 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 $exp\left\{-\text{const}·{\mathit{t}}^{1/3}\right\}$ for $\mathit{u}\left(\mathit{t},\mathit{x}\right)$ when x lies in a valley. We demonstrate also that the length of a valley grows at least as $exp\left\{\mathbf{+}\text{const}·{\mathit{t}}^{1/3}\right\}$ as $\mathit{t}\to \infty$.

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

## Funding Statement

The first author was supported in part by the National Science Foundation grant DMS-1855439. The second author was supported by the National Research Foundation of Korea grants 2019R1A5A1028324 and RS-2023-00244382. The third author was supported by Simons Foundation Collaboration Grant 513424.

## Citation

Davar Khoshnevisan. Kunwoo Kim. Carl Mueller. "On the valleys of the stochastic heat equation." Ann. Appl. Probab. 34 (1B) 1177 - 1198, February 2024. https://doi.org/10.1214/23-AAP1988

## Information

Received: 1 November 2022; Revised: 1 May 2023; Published: February 2024
First available in Project Euclid: 1 February 2024

MathSciNet: MR4700256
Digital Object Identifier: 10.1214/23-AAP1988

Subjects:
Primary: 60H15
Secondary: 35K05 , 35R60

Keywords: dissipation , Parabolic Anderson model , The stochastic heat equation , valleys