Symmetrization of exterior parabolic problems and probabilistic interpretation

TL;DR

This paper establishes a comparison theorem for solutions of exterior parabolic problems and offers a PDE-based proof of the isoperimetric inequality for the Wiener sausage through symmetrization techniques.

Contribution

It introduces a PDE-based comparison theorem for symmetrized exterior parabolic problems using polarization approximations.

Findings

Comparison theorem for averages of solutions to symmetrized exterior parabolic problems

Alternative PDE proof of the isoperimetric inequality for the Wiener sausage

Use of polarization to approximate Schwarz symmetrization

Abstract

We prove a comparison theorem for the averages of the solutions of two exterior parabolic problems, the second being the "symmetrization" of the first one, by using approximation of the Schwarz symmetrization by polarizations, as it was introduced in [4]. This comparison provides an alternative proof, based on PDEs, of the isoperimetric inequality for the Wiener sausage, which was proved in [14].