Time regularity for generalized Mehler semigroups

Abstract

We study continuity and Hölder continuity of 𝑡 ↦ 𝑃 𝑡 𝑓, where 𝑃 𝑡 is a generalized Mehler semigroup in 𝐶 𝑏 (𝑋), the space of the continuous and bounded functions from a Banach space 𝑋 to ℝ, and 𝑓 ∈ 𝐶 𝑏 (𝑋). The generators 𝐿 of such semigroups are realizations of a class of differential and pseudo-differential operators, both in finite and in infinite dimension. Examples of operators 𝐿 to which this theory is applicable include Ornstein-Uhlenbeck operators with fractional diffusion in finite dimension, and Ornstein-Uhlenbeck operators with associated strong-Feller semigroups, in infinite dimension.