AI 中文总结
研究高斯空间上\(\mathbb{R}^k\)值Sobolev映射律间总变差距离，利用Malliavin行列式小球界，通过Kantorovich - Rubinstein距离等估计，基于分数正则性估计证明，为特定随机向量分布提供新估计及更高指数。
AI 中文摘要
在高斯空间上两个\(\mathbb{R}^k\)值Sobolev映射的 Malliavin 行列式的小球界条件下，我们根据 Kantorovich - Rubinstein 距离以及相应 Sobolev 空间中映射之间的距离来估计它们的律之间的总变差距离。特别地，我们的结果为其分量属于 Wiener 混沌有限和的随机向量分布提供了新的总变差距离估计，指数提高了一个渐近因子二。证明基于 Sobolev 映射分布的分数正则性估计。即，我们表明如果一个\(\mathbb{R}^k\)值映射的分量在\(W^{2,p}(\gamma)\)中，并且相应 Malliavin 矩阵的行列式满足阶数为\(\varkappa\in(0,1]\)的小球界，那么该映射的律具有阶数为\(\frac{\varkappa}{1+(2k - 1)\varkappa p^{-1}}\)的分数正则性。特别地，对于大\(p\)，这给出了阶数为\(\varkappa\)且有\(O(p^{-1})\)损失的正则性。
英文摘要
Under small-ball bounds for the Malliavin determinants of two $\mathbb R^k$-valued Sobolev mappings on a Gaussian space, we estimate the total variation distance between their laws both in terms of the Kantorovich--Rubinstein distance and in terms of the distance between the mappings in the corresponding Sobolev space. In particular, our results yield new total variation distance estimates for distributions of random vectors whose components belong to finite sums of Wiener chaoses, with exponents improved by an asymptotic factor of two. The proof is based on fractional regularity estimates for distributions of Sobolev mappings. Namely, we show that if an $\mathbb R^k$-valued mapping has components in $W^{2,p}(γ)$ and the determinant of the corresponding Malliavin matrix satisfies a small-ball bound of order $\varkappa\in(0,1]$, then the law of the mapping has fractional regularity of order \[ \frac{\varkappa}{1+(2k-1)\varkappa p^{-1}}. \] In particular, for large $p$, this gives regularity of order $\varkappa$ up to an $O(p^{-1})$ loss.
