发表机构
Technion—Israel Institute of Technology（以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明三阶Hermite过程的Malliavin非退化性，通过无限秩高斯二次型和Fourier界获得负矩与密度光滑性，衰减指数为2/3。
AI 中文摘要
我们证明了三阶Hermite过程的Malliavin非退化性。关键步骤是证明，对于每个非零测试方向$h\in C_c^\infty(0,1)$，三阶Hermite随机变量的方向Malliavin导数是一个无限秩的高斯二次型。利用此类方向的任意大的标准正交族，我们推导出其联合特征函数的自包含Fourier界，并获得Malliavin范数的任意阶多项式小球估计。这给出了每个阶的负矩。将单次估计与行列式分解及Malliavin强局部非确定性相结合，我们获得了有限维Malliavin行列式的所有阶负矩。因此，所有有限维分布以及非重叠增量的任意向量都承认Schwartz密度。我们进一步建立了归一化增量向量的逆Malliavin行列式的网格一致Sobolev界，并推导出其密度的所有偏导数的拉伸指数估计。衰减指数为$2/3$，反映了第三Wiener混沌。这解决了Rosenblatt情形之后的下一个非高斯Hermite阶，并为二阶光滑性理论提供了定量对应。
英文摘要
We prove Malliavin nondegeneracy for the third-order Hermite process. The key step is to show that, for every nonzero test direction $h\in C_c^\infty(0,1)$, the directional Malliavin derivative of a third-order Hermite random variable is an infinite-rank Gaussian quadratic form. Using arbitrarily large orthonormal families of such directions, we derive a self-contained Fourier bound for their joint characteristic function and obtain polynomial small-ball estimates of arbitrary order for the Malliavin norm. This yields negative moments of every order. Combining the one-time estimate with determinant factorization and Malliavin strong local nondeterminism, we obtain negative moments of all orders for finite-dimensional Malliavin determinants. Consequently, all finite-dimensional distributions, as well as arbitrary vectors of non-overlapping increments, admit Schwartz densities. We further establish grid-uniform Sobolev bounds for inverse Malliavin determinants of normalized increment vectors and derive stretched-exponential estimates for all partial derivatives of their densities. The decay exponent is $2/3$, reflecting the third Wiener chaos. This settles the next non-Gaussian Hermite order after the Rosenblatt case and provides a quantitative counterpart to the order-two smoothness theory.
