Symmetrization of L\'evy processes and applications

TL;DR

This paper extends classical isoperimetric inequalities from Brownian motion to a broad class of Levy processes, utilizing probabilistic structures and integral inequalities for proofs.

Contribution

It introduces a framework for applying isoperimetric inequalities to Levy processes, broadening the scope beyond Brownian motion with novel probabilistic methods.

Findings

Classical inequalities extend to Levy processes

Probabilistic structure is crucial for proofs

Multiple integral inequalities underpin the results

Abstract

It is shown that many of the classical generalized isoperimetric inequalities for the Laplacian when viewed in terms of Brownian motion extend to a wide class of Levy processes. The results are derived from the multiple integral inequalities of Brascamp, Lieb and Luttinger but the probabilistic structure of the processes plays a crucial role in the proofs.