On the valleys of the stochastic heat equation

TL;DR

This paper investigates the properties of valleys in solutions to the stochastic heat equation driven by space-time white noise, revealing how valley size and length evolve over time and under certain initial conditions.

Contribution

It provides new theoretical results on the asymptotic behavior of valleys, including bounds on their size and length, and conditions for infinite valley length.

Findings

Supremum over valleys vanishes as time increases, with a specific decay rate.

Valley length grows at least exponentially with time.

Valleys become infinitely long for subgaussian initial data.

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. High peaks of solutions have been extensively studied under the name of intermittency, but less is known about spatial regions between peaks, which 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 over a valley. More precisely, we show that the supremum of the solution over a valley vanishes as $t\to\infty$, and we establish an upper bound of $\exp\{-\text{const}\cdot t^{1/3}\}$ for the rate of decay. We demonstrate also that the length of a valley grows at least as $\exp\{+\text{const}\cdot t^{1/3}\}$ as $t\to\infty$. Our second theorem asserts that the length of the valleys are eventually infinite when the initial data has subgaussian tails.