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

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 related stochastic convolution. then introduce corresponding Ornstein–Uhlenbeck focusing on regularizing properties Markov transition semigroup defined by it. In particular, we provide explicit, original formula –which is not Bismut–Elworthy–Li's type– for Gateaux derivatives functions generated operators semigroup, as well upper bound norm their gradients. case α∈(12,1), this estimate represents starting point studying Kolmogorov equation its mild formulation.