Probability Seminar: Carl Mueller, University of Rochester

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VALLEYS OF THE STOCHASTIC HEAT EQUATION

This is joint work with Mathew Joseph, Kunwoo Kim, and Davar Khoshnevisan.

Intermittency is a well-studied property of the stochastic heat equation. Namely, for large time, the solution is dominated by tall peaks separated by large valleys where the solution is small. The behavior of the peaks has attracted the most attention, and not many results deal with the valleys.

In earlier work, Kunwoo Kim, Davar Khoshnevisan and I considered valleys of the stochastic heat equation in on the real line, with a nonlinear term multiplying white noise. We showed that as time t grows, the valleys become wider at least at an exponential rate with exponent a constant times t to the 1/3. Also, the supremum of the solution within the valley vanishes at least at an exponential rate, with exponent a constant times t to the -1/3.

Experts believed that the 1/3 which appears in the exponentials could be replaced by 1. But due to the nonlinear coefficient multiplying the noise, integrable probability cannot be applied. With the help of Mathew Joseph, we show that 1/3 can indeed be replaced by 1. We also extend this result to higher dimensions, with white noise replaced by colored noise that is sufficiently strong in a way that we will explain.