论文标题
在随机热方程的山谷上
On the valleys of the stochastic heat equation
论文作者
论文摘要
我们考虑了由时空白噪声（也称为随机热方程）在实际线上驱动的抛物线安德森模型的概括。溶液的高峰已以间歇性的名义进行了广泛的研究，但对峰之间的空间区域知之甚少，峰之间的空间区域可能会松散地称为山谷。我们提供了有关溶液谷的两个结果。 我们的第一个定理提供了有关山谷大小和山谷上解决方案的最高信息的信息。更准确地说，我们表明，山谷上的解决方案的上限消失为$ t \ to \ infty $，并且我们建立了$ \ exp \ { - \ text { - \ text {const} \ cdot t^{1/3} \} $的上限。我们还证明，山谷的长度至少增长为$ \ exp \ {+\ text {const} \ cdot t^{1/3} \} $作为$ t \ to \ infty $。 我们的第二个定理断言，当初始数据具有Subgaussian尾巴时，山谷的长度最终是无限的。
We consider a generalization of the parabolic Anderson model driven by space-time white noise, also called the stochastic heat equation, on the real line. High peaks of solutions have been extensively studied under the name of intermittency, but less is known about spatial regions between peaks, which may loosely refer to as valleys. We present two results about the valleys of the solution. Our first theorem provides information about the size of valleys and the supremum of the solution over a valley. More precisely, we show that the supremum of the solution over a valley vanishes as $t\to\infty$, and we establish an upper bound of $\exp\{-\text{const}\cdot t^{1/3}\}$ for the rate of decay. We demonstrate also that the length of a valley grows at least as $\exp\{+\text{const}\cdot t^{1/3}\}$ as $t\to\infty$. Our second theorem asserts that the length of the valleys are eventually infinite when the initial data has subgaussian tails.
