Gaugeability and conditional gaugeability

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.