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Symmetrization, symmetric stable processes, and Riesz capacities

Author: Dimitrios Betsakos
Journal: Trans. Amer. Math. Soc. 356 (2004), 735-755
MSC (2000): Primary 31B15, 60J45
Published electronically: September 22, 2003
Addendum: Trans. Amer. Math. Soc. (recently posted)
MathSciNet review: 2022718
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Abstract: Let $\texttt{X}_t$ be a symmetric $\alpha$-stable process killed on exiting an open subset $D$ of $\mathbb R^n$. 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 $B$ in the complement of $D$ in the first exit moment from $D$ increases when $D$ and $B$ 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 $K$ in $\mathbb R^n$with given volume, the balls have the least $\alpha$-capacity ( $0<\alpha<2$).

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Additional Information

Dimitrios Betsakos
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Keywords: Symmetrization, symmetric stable process, polarization, transition function, $\alpha$-harmonic measure, Green function, Riesz capacity
Received by editor(s): July 14, 2002
Received by editor(s) in revised form: January 23, 2003
Published electronically: September 22, 2003
Dedicated: Dedicated to Albert Baernstein on the occasion of the thirty years of his star-function
Article copyright: © Copyright 2003 American Mathematical Society

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