On Steiner Symmetrizations of First Exit Time Distributions and Lévy Processes

Timothy Rolling, Purdue University

Abstract

The goal of this thesis is to establish generalized isoperimetric inequalities on first exit time distributions as well as expectations of Lévy processes.Firstly, we prove inequalities on first exit time distributions in the case that the Lévy process is an α-stable symmetric process At on R d , α P p0, 2s. Given At and a bounded domain D Ă Rd , we present a proof, based on the classical Brascamp-Lieb-Luttinger inequalities for multiple integrals, that the distribution of the first exit time of At from D increases under Steiner symmetrization. Further, it is shown that when a sequence of domains tDmu each contained in a ball B Ă R d and satisfying the ε-cone property converges to a domain D1 with respect to the Hausdorff metric, the sequence of distributions of first exit times for Brownian motion from Dm converges to the distribution of the exit time of Brownian motion from D1. The second set of results in this thesis extends the theorems from [5] by proving generalized isoperimetric inequalities on expectations of Lévy processes in the case of Steiner symmetrization.These results will then be used to establish inequalities involving distributions of first exit times of α-stable symmetric processes Atfrom triangles and quadrilaterals. The primary application of these inequalities is verifying a conjecture from Bañuelos in [3] for these planar domains. This extends a classical result of Pólya and Szegö in [28] to the fractional Laplacian with Dirichlet boundary conditions.

Degree

Ph.D.

Advisors

Bañuelos, Purdue University.

Subject Area

Mathematics

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