Malliavin smoothness of the Rosenblatt process

TL;DR

This paper studies the smoothness of the Rosenblatt process's distributions using Malliavin calculus, proving nondegeneracy and rapid decay of densities, with implications for understanding its probabilistic structure.

Contribution

It establishes the nondegeneracy of Rosenblatt process vectors in Malliavin calculus and characterizes their densities as Schwartz functions, providing new insights into their smoothness properties.

Findings

Rosenblatt process densities are in the Schwartz space.

All negative moments of the Malliavin matrix determinant exist.

Derived exponential bounds for density derivatives.

Abstract

We investigate the smoothness of the densities of the finite-dimensional distributions of the Rosenblatt process. Within the Malliavin calculus framework, we prove that Rosenblatt random vectors are nondegenerate in the Malliavin sense. As a consequence, their densities belong to the Schwartz space of rapidly decreasing smooth functions. The proof relies on establishing the existence of all negative moments of the determinant of the Malliavin matrix, exploiting the specific structure of random variables in the second Wiener chaos. In addition, we derive exponential-type upper bounds for the partial derivatives of the densities of the finite-dimensional distributions of the Rosenblatt process.