Smoothing effect and Derivative formulas for Ornstein-Uhlenbeck processes driven by subordinated cylindrical Brownian noises

TL;DR

This paper studies Ornstein-Uhlenbeck processes driven by subordinated cylindrical Brownian noises in infinite dimensions, providing explicit derivative formulas and regularity results for the associated semigroup, with implications for the Kolmogorov equation.

Contribution

It introduces a novel explicit formula for Gateaux derivatives of the semigroup functions, not of Bismut-Elworthy-Li type, and analyzes regularizing properties of the process.

Findings

Derived explicit formulas for derivatives of the semigroup functions.

Established upper bounds for the gradients of these derivatives.

Provided foundational results for studying the Kolmogorov equation in this context.

Abstract

We investigate the concept of cylindrical Wiener process subordinated to a strictly $\alpha$-stable L\'evy process, with $\alpha\in\left(0,1\right)$, 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 $\alpha\in\left(\frac{1}{2},1\right)$, this estimate represents the starting point for studying the Kolmogorov equation in its mild formulation.