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Malliavin smoothness and density estimates for the third-order Hermite process — arXiv:2609.07957

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.07957
License (as archived): CC0-1.0
Archived: 2026-10-16T14:32:38.288Z
SHA-256 of the archived text: 84963886ab87017be0d1eaee46c3cc04f420cd8f262ef385560b61ae26d7b1e7
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How to cite

Malliavin smoothness and density estimates for the third-order Hermite process — arXiv:2609.07957 — arXiv (en), https://arxiv.org/abs/2609.07957 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_2d3385eef97d (Metraipolis archive dc_2d3385eef97d, sha256 84963886ab87017be0d1eaee46c3cc04f420cd8f262ef385560b61ae26d7b1e7)

BibTeX

@misc{dc2d3385eef97d,
  title = {Malliavin smoothness and density estimates for the third-order Hermite process — arXiv:2609.07957},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.07957},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_2d3385eef97d; sha256 84963886ab87017be0d1eaee46c3cc04f420cd8f262ef385560b61ae26d7b1e7},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.07957. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-09 in math.PR).
Categorie arXiv: Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2609.07957 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T14:09:59.783Z dalla pagina https://arxiv.org/catchup/math.PR/2026-09-09?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).