The sample paths of white noise are proved to be elements of certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse with increasing space dimension. Consequently, white noise is actually much smoother than the known sharp regularity results in isotropic spaces suggest. An application of our techniques yields new results for the regularity of solutions of Poisson and heat equation on the half space with boundary noise. The main novelty is the flexible treatment of the interplay between the singularity at the boundary and the smoothness in tangential, normal and time direction.

中文翻译：

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具有主导混合平滑度的空间中白噪声的采样路径

白噪声的样本路径被证明是某些 Besov 空间的元素，具有主导混合平滑度。与各向同性空间不同，这里的规律性不会随着空间维度的增加而变差。因此，白噪声实际上比各向同性空间中已知的尖锐规律性结果要平滑得多。我们的技术的应用产生了关于具有边界噪声的半空间上泊松和热方程解的规律性的新结果。主要新颖之处在于灵活处理边界处的奇点与切线、法线和时间方向的平滑度之间的相互作用。