Limit theorems for the generator of a symmetric Levy process with the delta potential

TL;DR

This paper develops a mathematical framework for symmetric Levy processes with delta potentials, extending the Feynman-Kac formula and establishing limit theorems for the process and its sample paths.

Contribution

It constructs a self-adjoint extension of the generator incorporating delta potentials and proves related limit theorems, advancing the understanding of Levy processes with singular interactions.

Findings

Extended Feynman-Kac formula for delta potentials

Proved limit theorems for operator semigroups

Constructed distributions attracting sample paths

Abstract

We consider a one-dimensional symmetric Levy process that has local time. In the first part, we construct a self-adjoint extension of the generator of the process so that the constructed operator corresponds to the generator with the delta potential. Using the constructed operator, we extend the Feynman-Kac formula to the case of delta function-type potentials and prove a limit theorem for an operator semigroup corresponding to this formula. In the second part, we construct a one-parameter family of distributions that attract the sample paths of the process to a given point. We show that this family weakly converges to the distribution of a Feller process and prove a limit theorem for the distribution of a point where an attracted sample path comes.