We study regularizing properties for transition semigroups related to Ornstein Uhlenbeck processes with values in a Banach space $E$ which is continuously and densely embedded in a real and separable Hilbert space $H$. Namely we study conditions under which the transition semigroup maps continuous and bounded functions into differentiable functions. Via a Girsanov type theorem such properties extend to perturbed Ornstein Uhlenbeck processes. We apply the results to solve in mild sense semilinear versions of Kolmogorov equations in $E$.
"Regularizing Properties for Transition Semigroups and Semilinear Parabolic Equations in Banach Spaces." Electron. J. Probab. 12 387 - 419, 2007. https://doi.org/10.1214/EJP.v12-401