Dissipation in parabolic SPDEs

TL;DR

This paper investigates the behavior of solutions to parabolic SPDEs on finite intervals, revealing that solutions tend to zero exponentially fast almost surely, especially under large noise conditions.

Contribution

It establishes the almost sure exponential decay of the spatial supremum of solutions to parabolic SPDEs on finite intervals, focusing on the near-zero probability set.

Findings

Solutions' supremum tends to zero exponentially fast over time.

High noise increases the probability of solutions remaining near zero.

The study extends intermittency analysis to the near-zero regime.

Abstract

The study of intermittency for the parabolic Anderson problem usually focuses on the moments of the solution which can describe the high peaks in the probability space. In this paper we set up the equation on a finite spatial interval, and study the other part of intermittency, i.e., the part of the probability space on which the solution is close to zero. This set has probability very close to one, and we show that on this set, the supremum of the solution over space is close to 0. As a consequence, we find that almost surely the spatial supremum of the solution tends to zero exponentially fast as time increases. We also show that if the noise term is very large, then the probability of the set on which the supremum of the solution is very small has a very high probability.