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Transactions of the American Mathematical Society

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Gaugeability and conditional gaugeability

Author: Zhen-Qing Chen
Journal: Trans. Amer. Math. Soc. 354 (2002), 4639-4679
MSC (2000): Primary 60J45, 60J57; Secondary 35J10, 35S05, 47J20, 60J35
Published electronically: July 2, 2002
MathSciNet review: 1926893
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Abstract: New Kato classes are introduced for general transient Borel right processes, for which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Green-tight measures in the classical Brownian motion case. However, the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability. We show that gaugeability, conditional gaugeability and the subcriticality for the associated Schrödinger operators are equivalent for transient Borel right processes with strong duals. Analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric Markov processes. These analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases. Connections with the positivity of the spectral radii of the associated Schrödinger operators are also established.

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Additional Information

Zhen-Qing Chen
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195

Keywords: Green function, $h$-transform, conditional Markov process, lifetime, time change, Kato class, Feynman-Kac transform, Schr\"odinger semigroup, Stieltjes exponential, non-local perturbation, spectral radius, gauge theorem, conditional gauge theorem, super gauge theorem, super conditional gauge theorem, subcriticality, bilinear form
Received by editor(s): August 12, 2001
Received by editor(s) in revised form: February 7, 2002
Published electronically: July 2, 2002
Additional Notes: The research of this author is supported in part by NSF Grant DMS-0071486
Article copyright: © Copyright 2002 American Mathematical Society