# Lecture 13

with this lecture we enter into the topic of elliptic operators with unbounded coefficients, analyzing in details the Ornstein-Uhlenbeck operator which is the prototype of an elliptic operator with unbounded coefficients. For the Ornstein-Uhlenbeck semigroup things are easier since (as in the case of the Laplacian) an explicit formula is available for the associated semigroup and this greatly simplifies the analysis of the semigroup and the nonhomogeneous Cauchy problem associated with the O-U operator. Differently from the case of bounded coefficients, here we can prove optimal Schauder estimates for solutions to the nonhomogeneous Cauchy problems associated with O-U, only with respect to the spatial variables: a typical feature of the case of unbounded coefficients. We also mention (and we will show it in the lecture) that the O-U semigroup is neither strongly continuous nor analytic in the space of bounded and (uniformly) continuous functions over $\mathbb R^d$.