A Feynman-Kac-It\^o Formula for magnetic Schr\"odinger operators on graphs

Abstract

In this paper we prove a Feynman-Kac-It\^{o} formula for magnetic Schr\"odinger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schr\"odinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman-Kac-It\^{o} formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman-Kac-It\^{o} formula, we prove a Kato inequality, a Golden-Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials.