Uniform integrability of exponential martingales and spectral bounds of non-local Feynman-Kac semigroups

TL;DR

This paper introduces a new, easily verifiable criterion for the uniform integrability of exponential martingales in Markov processes and applies it to analyze spectral bounds of Feynman-Kac semigroups, including non-local cases.

Contribution

It provides a novel criterion for uniform integrability of exponential martingales and extends spectral bound analysis to non-local Feynman-Kac semigroups using Girsanov transforms.

Findings

New criterion for uniform integrability of exponential martingales.

Criteria for $L^p$-independence of spectral bounds.

Extension of spectral analysis to non-local Feynman-Kac semigroups.

Abstract

In the first part of this paper, we give a useful criterion for uniform integrability of exponential martingales in the context of Markov processes. The condition of this criterion is easy to verify and is, in general, much weaker than the commonly used Novikov's condition. In the second part of this paper, we present a new approach to the study of spectral bounds of Feynman-Kac semigroups for a large class of symmetric Markov processes. We first establish criteria for the $L^p$-independence of spectral bounds for Feynman-Kac semigroups generated by continuous additive functionals, using gaugeability results obtained by the author in \cite{C}. We then extend these analytic criteria for the $L^p$-independence of spectral bounds to non-local Feynman-Kac semigroups via pure jump Girsanov transforms. For this, the uniform integrability of the exponential martingales established in the first part of this paper plays an important role. We use it to show that Kato classes introduced in \cite{C} can only become larger under pure jump Girsanov transforms with symmetric jumping functions.