论文标题
山峰的景观：随机热方程的间歇岛和lévy噪声
A landscape of peaks: The intermittency islands of the stochastic heat equation with Lévy noise
论文作者
论文摘要
我们表明，如果驱动噪声是非高斯，那么溶液对随机热方程的空间曲线具有多层间歇岛。一方面，正如预期的那样，如果噪声足够重，则溶液的最大峰在乘法下将比添加噪声高。另一方面，令人惊讶的是，一旦噪声具有有限的顺序$ \ frac2d $，其中$ d $是空间尺寸，最大的峰将是相同的添加剂和乘法噪声的顺序，与高斯噪声下的溶液的行为形成了鲜明的对比。但是，在这种情况下，仔细检查揭示了第二层的峰，在最大峰之下，这是乘法噪声独有的，可以通过在晶格上抽样溶液来观察。最后，我们计算溶液间歇岛的宏观Hausdorff和Minkowski尺寸。在添加噪声和乘法噪声下，如果不是太重，最大的峰将是自相似的，就其大规模的多重绘制行为而言。但是在乘法噪声下，这种类型的自相似性在晶格上观察到的峰中不存在。
We show that the spatial profile of the solution to the stochastic heat equation features multiple layers of intermittency islands if the driving noise is non-Gaussian. On the one hand, as expected, if the noise is sufficiently heavy-tailed, the largest peaks of the solution will be taller under multiplicative than under additive noise. On the other hand, surprisingly, as soon as the noise has a finite moment of order $\frac2d$, where $d$ is the spatial dimension, the largest peaks will be of the same order for both additive and multiplicative noise, which is in sharp contrast to the behavior of the solution under Gaussian noise. However, in this case, a closer inspection reveals a second layer of peaks, beneath the largest peaks, that is exclusive to multiplicative noise and that can be observed by sampling the solution on the lattice. Finally, we compute the macroscopic Hausdorff and Minkowski dimensions of the intermittency islands of the solution. Under both additive and multiplicative noise, if it is not too heavy-tailed, the largest peaks will be self-similar in terms of their large-scale multifractal behavior. But under multiplicative noise, this type of self-similarity is not present in the peaks observed on the lattice.
