Symmetric rearrangements around infinity with applications to Lévy processes

Abstract

We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger (Arch Ration Mech 61:35-44, 1976) and can be interpreted as involving symmetric rearrangements of domains around ∞. As applications, we prove two comparison results for general Lévy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Lévy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Lévy motions are symmetrically rearranged. This essentially generalizes an isoperimetric inequality of Peres and Sousi (Geom Funct Anal 22(4):1000-1014, 2012) for the Wiener sausage. In the second application, we show that the q-capacity of a Borel measurable set for a Lévy process can only decrease if the set and the Lévy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe (Z Wahrsch Verw Gebiete 63:487-499, 1983) for symmetric Lévy processes. © 2013 Springer-Verlag Berlin Heidelberg.