论文标题
小球概率，公制熵和高斯粗糙路径
Small ball probabilities, metric entropy and Gaussian rough paths
论文作者
论文摘要
我们研究高斯粗糙路径的小球概率（SBP）。尽管许多在粗糙路径上研究了由高斯粗糙路径驱动的随机过程的大偏差原理（LDP），但在文献中，SBP并未扩展到粗糙的路径框架。 LDP提供了有关建立可集成性类型属性的度量的宏观信息。 SBP提供了微观信息，用于建立局部准确的近似值。鉴于繁殖的内核希尔伯特空间（RKHS）球的紧凑性，其度量熵提供了有关如何近似高斯粗糙路径定律的宝贵信息。 作为一种应用，我们能够找到上限和下限，以实现经验性粗糙的高斯度量与其在路径空间中的真实定律的收敛速度。
We study the Small Ball Probabilities (SBPs) of Gaussian rough paths. While many works on rough paths study the Large Deviations Principles (LDPs) for stochastic processes driven by Gaussian rough paths, it is a noticeable gap in the literature that SBPs have not been extended to the rough path framework. LDPs provide macroscopic information about a measure for establishing Integrability type properties. SBPs provide microscopic information and are used to establish a locally accurate approximation for a measure. Given the compactness of a Reproducing Kernel Hilbert space (RKHS) ball, its metric entropy provides invaluable information on how to approximate the law of a Gaussian rough path. As an application, we are able to find upper and lower bounds for the rate of convergence of an empirical rough Gaussian measure to its true law in pathspace.
