Conditional gauge theorem for non-local Feynman-Kac transforms

Abstract

Abstract. Feynman-Kac transforms driven by discontinuous additive functionals are studied in this paper for a large class of Markov processes. General gauge and conditional gauge theorems are established for such transforms. Furthermore, the L 2 -infinitesimal generator of the Schrödinger semigroup given by a non-local Feynman-Kac transform is determined in terms of its associated bilinear form.

Perturbation of symmetric Markov processes

“…In our recent paper [3], we established the needed stochastic integration theory for zero-energy additive functionals of X as well as the corresponding Itô formula via time-reversal technique. The main result of the current paper extends not only the results in [14] and [9] but also Feynman-Kac transforms by continuous additive functionals of zero energy studied in Chen and Zhang [8] and the pure-jump Girsanov transforms and discontinuous Feynman-Kac transforms in Chen [1] and in Chen and Song [5]- [6]. The following is a more detailed description of this paper.…”

“…See section 2 below for a quick review. A special case ( M = 0 and M purely discontinuous) of the above result was obtained by Chen and Song [5] (see also [1] and [6]) in the broader context of "nearly symmetric" right Markov processes, under somewhat more stringent conditions on µ M and µ (in [5] µ M is assumed to be in the Kato class while µ is only assumed to satisfy the condition Gµ + ∞ < 1). Recall that the Revuz measure ν of a PCAF A ν is said to be of the Kato class provided…”

Estimates on the Transition Densities of Girsanov Transforms of Symmetric Stable Processes

“…This theorem is a partial generalization of Theorem 4.8 in Ref. 7, where F was assumed to be in J(X), while we only assume that F ∈ I 2 (X).…”

Estimates and structure of α-harmonic functions

“…Technically, the development hinges on Lemma 7 and Lemma 8 below, and extensions of these should be sought for in the more general settings. A certain role is also played by (22).…”

“…The role of BHP in estimating the Green function and studying Schrödinger-type operators is also well understood. For more information on typical applications we refer the reader to [11], [30], [12], [20], [13], [14], [22], see also [25] for a general perspective.…”