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On the valleys of the stochastic heat equation

https://doi.org/10.1214/23-AAP1988

Khoshnevisan, Davar; Kim, Kunwoo; Mueller, Carl (February 2024, The Annals of Applied Probability)

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Small-ball constants, and exceptional flat points of SPDEs

https://doi.org/10.1214/24-EJP1242

Khoshnevisan, Davar; Kim, Kunwoo; Mueller, Carl (January 2024, Electronic Journal of Probability)

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Dissipation in Parabolic SPDEs

https://doi.org/10.1007/s10955-020-02540-0

Khoshnevisan, Davar; Kim, Kunwoo; Mueller, Carl; Shiu, Shang-Yuan (April 2020, Journal of Statistical Physics)

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Dissipation in parabolic SPDEs

Khoshnevisan, Davar; Kim, Kunwoo; Mueller, Carl; Shiu, Shang-Yuan (January 2020, Journal of statistical physics)

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.
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