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.

中文翻译：

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抛物线 SPDE 的耗散

抛物线安德森问题的间歇性研究通常集中在可以描述概率空间中高峰的解的矩上。本文在有限空间区间上建立方程，研究间歇性的另一部分，即概率空间中解接近于零的部分。这个集合的概率非常接近 1，我们证明在这个集合上，空间上的解的上限值接近 0。因此，我们发现几乎可以肯定，解的空间上限值以指数方式快速趋于零，如下所示时间增加。我们还表明，如果噪声项非常大，则解的上限值非常小的集合的概率非常高。