Cameron--Martin Type Theorem for a Class of non-Gaussian Measures

TL;DR

This paper extends the Cameron--Martin theorem to a class of non-Gaussian measures linked to generalized grey Brownian motions, identifying their Cameron--Martin space and deriving explicit Radon-Nikodym densities.

Contribution

It introduces a Cameron--Martin type theorem for non-Gaussian measures and provides explicit formulas for Radon-Nikodym derivatives in this context.

Findings

Identified the Cameron--Martin space for the measures

Derived explicit Radon-Nikodym densities

Established integration by parts formula

Abstract

In this paper, we study the quasi-invariant property of a class of non-Gaussian measures. These measures are associated with the family of generalized grey Brownian motions. We identify the Cameron--Martin space and derive the explicit Radon-Nikodym density in terms of the Wiener integral with respect to the fractional Brownian motion. Moreover, we show an integration by parts formula for the derivative operator in the directions of the Cameron--Martin space. As a consequence, we derive the closability of both the derivative and the corresponding gradient operators.