Smoothing effect and derivative formulas for Ornstein–Uhlenbeck processes driven by subordinated cylindrical Brownian noises

Abstract

We investigate the concept of cylindrical Wiener process subordinated to a strictly α–stable Lévy process, with α∈(0,1), in an infinite–dimensional, separable Hilbert space, and consider the related stochastic convolution. We then introduce the corresponding Ornstein–Uhlenbeck process, focusing on the regularizing properties of the Markov transition semigroup defined by it. In particular, we provide an explicit, original formula –which is not of Bismut–Elworthy–Li's type– for the Gateaux derivatives of the functions generated by the operators of the semigroup, as well as an upper bound for the norm of their gradients. In the case α∈(12,1), this estimate represents the starting point for studying the Kolmogorov equation in its mild formulation.