Malliavin differentiability for the excursion measure of a Gaussian field

TL;DR

This paper studies the Malliavin differentiability of the excursion measure of Gaussian fields, establishing conditions based on covariance decay for continuous-index fields, with implications for understanding their regularity.

Contribution

It provides a precise characterization of when the excursion measure of a Gaussian field is Malliavin differentiable in terms of covariance decay rates, extending the theory to continuous-index fields.

Findings

Identifies polynomial decay conditions on covariance for Malliavin differentiability.

Shows the approach applies to stationary and isotropic Gaussian fields on Rd and Sd.

Provides explicit criteria linking covariance decay to regularity of excursion measures.

Abstract

In this work, we investigate the Malliavin differentiability of the excursion volume of a Gaussian field. In particular, we prove that if the moments of the covariance kernel go to 0 polinomially, then we can determine exactly for which p the excursion measure has pth square integrable Malliavin derivative. While our approach can not be implemented for Gaussian fields indexed by discrete sets (where the moments do not vanish), when the Gaussian field is indexed by Rd and stationary, or indexed by Sd and isotropic, the condition on the polynomial decay of the mentioned moments can be inferred starting from simple and general conditions on the covariance function of the field.