Symmetric rearrangements around infinity with applications to L,vy processes

• 成果类型: Article
• 署名作者: Drewitz, Alexander; Sousi, Perla; Sun, Rongfeng
• 署名单位: Columbia University; University of Cambridge; National University of Singapore
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0492-1
• 发表日期: 2014
• 页码: 637-664
• 关键词: isoperimetric inequality; riesz capacities; Levy processes

摘要

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 L,vy 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 L,vy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the L,vy 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 -capacity of a Borel measurable set for a L,vy process can only decrease if the set and the L,vy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe (Z Wahrsch Verw Gebiete 63:487-499, 1983) for symmetric L,vy processes.