Malliavin Smoothness of Hermite Processes of Arbitrary Order and Their Wiener Integrals

Abstract

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.