Symmetrization, symmetric stable processes, and Riesz capacities

Abstract

Let X t \texttt {X}_t be a symmetric α \alpha -stable process killed on exiting an open subset D D of R n \mathbb R^n . We prove a theorem that describes the behavior of its transition probabilities under polarization. We show that this result implies that the probability of hitting a given set B B in the complement of D D in the first exit moment from D D increases when D D and B B are polarized. It can also lead to symmetrization theorems for hitting probabilities, Green functions, and Riesz capacities. One such theorem is the following: Among all compact sets K K in R n \mathbb R^n with given volume, the balls have the least α \alpha -capacity ( 0 > α > 2 0>\alpha >2 ).