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Problems on geometric properties of some solutions to the fractional Laplacian

Banff International Research Station for Mathematical Innovation and Discovery

A classical result of H.J Brascamp and E.H. Lieb says that the ground state eigenfunction for the Laplacian in convex domains (and of Schrödinger operators with convex potentials) on $\mathbb{R}^n$, is log-concave. A proof can be given (interpreted) in terms of the finite dimensional distributions of Brownian motion. Some years ago the speaker raised similar questions, and made some conjectures, when the Brownian motion is replaced by other stochastic processes and in particular by the rotationally symmetric $\alpha$-stable processes. These problems, for the most part, remain open even for an interval in $\mathbb{R}$. In this talk we elaborate on this topic, discuss some known results, and outline the proof of one of these.

Mathematics; Probability theory and stochastic processes; Functional analysis; Probability / statistical mechanics

video/mp4

Author affiliation: Purdue University

2017-06-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/62012

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/