General gauge theorem for multiplicative functionals

Abstract:

We generalize our previous work on the gauge theorem and its various consequences and complements, initiated in [8] and somewhat extended by subsequent investigations (see [6]). The generalization here is two-fold. First, instead of the Brownian motion, the underlying process is now a fairly broad class of Markov processes, not necessarily having continuous paths. Second, instead of the Feynman-Kac functional, the exponential of a general class of additive functionals is treated. The case of Schrödinger operator $\Delta /2 + \nu$, where $\nu$ is a suitable measure, is a simple special case. The most general operator, not necessarily a differential one, which may arise from our potential equations is briefly discussed toward the end of the paper. Concrete instances of applications in this case should be of great interest.