Functional Calculus via the extension technique
A talk in the Nonlocal Equations: Analysis and Numerics series by
David Lee from Sorbonne
| Abstract: | In this presentation, we present a solution to the problem:
"Which type of linear operators can be realized by the Dirichlet-to-Neumann operator associated with the operator $- \Delta - a(z)\frac{\partial^2}{\partial z^2}$ on an extension problem?'',
which was raised in the pioneering work [Comm. Par.Diff. Equ. 32 (2007)] by Caffarelli and Silvestre. We even intend to go a step further by replacing the negative Laplace operator $-\Delta$ on $\mathbb{R}^d$ by an $m$-accretive operator $A$ on a general Banach space $X$ and the Dirichlet-to-Neumann operator by the Dirichlet-to-Wentzell operator. We establish uniquness of solutions to the extension problem in the general Banach spaces framework, which seems to be new in the literature and of independent interest. We develop a type of functional calculus using probabilistic tools from excursion theory. With our method, we are able to characterize all linear operators $\psi(A)$, where $\psi$ is a complete Bernstein function, resulting in a new characterization of the famous Phillips’ subordination theorem within this class of functions.
This work was done jointly with Daniel Hauer from the University of Sydney. Within the CRC this talk is associated to the project(s): A7 |