Perturbations of L\'evy Processes Using the Feynman Functional

The Wiener process is an example of a stable L\'{e}vy process, and the stable L\'{e}vy processes have been used as a model for many Physical systems as well as Financial Mathematics. Using the heat semigroup, the Feynman-Kac formula provides a connection between the Wiener process and Feynman's path integral. DeFacio, Johnson, and Lapidus were able to expand upon this method to formulate a rigorous Feynman Functional for certain semigroup operators. It is possible to add a potential term to these operators in order to investigate the effect perturbations have on stable L\'{e}vy processes. With restrictions on the potential, these perturbations can be calculated using the methods developed for the Feynman path integral, and formal manipulation of the perturbations yields previous results in the literature. An overview of the Feynman Functional formulation for L\'{e}vy processes will be given, a few examples of perturbed L\'{e}vy processes will be shown, and some Mathematical and computational difficulties will be discussed.