Lecture 15

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Dear Participants,

with great regret we inform you that the last lecture of this year's internet-seminar can be now downloaded.

In Lecture 13 we have shown that the Ornstein-Uhlenbeck semigroup is not analytic in C_b(\mathbb R^d). In this lecture we present a large class of elliptic operators with unbounded drift term that generate non analytic semigroups on C_b(\R^d). For such class of operators we prove also that the imaginary axis is contained in the spectrum of the associated "weak generator". We provide also a sufficient condition implying generation of an analytic semigroup on C_b(\mathbb R^d).

Lecture15.pdf

This week we kindly ask the team from the Technische Universität Hamburg to prepare the official solutions to the exercises of the lecture.

In Appendix B (which has been updated also in the first part) we extend Theorem 13.2.1 to a class of more general elliptic operators with unbounded coefficients, emplowing the Bernstein method. For the sake of simplicity we confine ourselves to the case when the diffusion part of {\mathcal A} is the Laplacian but also more general operators can be considered (of course, under suitable assumptions on their coefficients!).

We will show more applications of the theory in the project phase, where we hope to see many of you participating. Details of the project phase and the workshop will be sent out in the following weeks.

Thank you again for participating, solving the exercises, and making comments in the Discussion Board. We enjoyed the first phase of the seminar very much.

Best wishes and have fun reading.

Your virtual lecturers.

Abdelaziz and Luca