Limit theorems for the generator of a symmetric Levy process with the delta potential
Temirlan Abildaev
TL;DR
The paper addresses limit theorems for the generator of a one-dimensional symmetric Lévy process with local time under a delta potential, $\mu\delta(x-a)$. It constructs a self-adjoint extension $\mathcal{A}_\mu$ of the original generator $\mathcal{A}$, establishes a delta-Feynman-Kac representation, and proves operator-semigroup limit theorems. It also builds a one-parameter family of path-attracting penalizations and proves convergence to a Feller process, including a detailed analysis of the associated transition density $\rho_\mu(t,x,y)$ and the invariant measure $\pi_\nu$. The results extend classical penalization and Feynman-Kac frameworks to delta-function-type potentials in the Lévy setting and provide a rigorous asymptotic description of attracted sample paths and endpoint distributions.
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.