任意阶Hermite过程及其Wiener积分的Malliavin光滑性

中文摘要 AI 辅助

我们针对奇异Hermite核上的每个加权积分建立了Malliavin非退化性，其Malliavin导数范数的所有负矩有限，且在有界确定性权重的容许紧族上一致成立。特别地，每个固定有限阶的Hermite过程的所有有限维分布具有Schwartz密度，非重叠增量向量以及更一般地，关于Hermite过程的有限族Wiener积分也是如此。对于此类向量，权重的线性无关性是绝对连续性的充分必要条件；在此条件下，所有逆Malliavin行列式矩存在，联合密度属于Schwartz空间，其界同样在紧族上一致。证明依赖于紧权重族上的非消失条件，该条件在方向微分下保持不变。方向分数变换的解析尾部性质在每个混沌水平验证了该条件。一致的Malliavin估计随后将低阶梯度界转化为任意多个一阶方向导数的有界联合密度，Bessel不等式完成归纳。该论证直接处理固定的非高斯律和奇异核，无需高斯极限假设，也无需单独局部非确定性转移。

英文摘要

We establish Malliavin nondegeneracy for every weighted integral against a singular Hermite kernel, with all negative moments of the Malliavin derivative norm finite, uniformly over admissible compact families of bounded deterministic weights. In particular, all finite-dimensional distributions of Hermite processes of every fixed finite order have Schwartz densities, as do vectors of non-overlapping increments and, more generally, finite families of Wiener integrals with respect to a Hermite process. For such vectors, linear independence of the weights is both necessary and sufficient for absolute continuity; under this condition, all inverse Malliavin determinant moments exist and the joint density belongs to the Schwartz space, with bounds that are again uniform over compact families. The proof rests on a nonvanishing condition on compact weight families that is preserved under directional differentiation. An analytic-tail property of fractional transforms of the directions verifies this condition at every chaos level. Uniform Malliavin estimates then convert lower-order gradient bounds into bounded joint densities of arbitrarily many first directional derivatives, and Bessel's inequality closes the induction. The argument works directly with fixed non-Gaussian laws and singular kernels, without a Gaussian-limit assumption and without a separate local-nondeterminism transfer.

发表机构

• Technion—Israel Institute of Technology（以色列理工学院）

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