Dissipation in parabolic SPDEs

Davar Khoshnevisan, Kunwoo Kim, Carl Mueller, Shang-Yuan Shiu

Published 2018-08-21Version 1

The study of intermittency for the parabolic Anderson problem usually focuses on the high peaks of the solution. In this paper we set up the equation on a finite spatial interval, and study 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 tends to 0 exponentially fast in time. As a consequence, we find that 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 large peaks decreases exponentially rapidly.