Large diffusion expansion of the transition kernel of the L\'evy Ornstein-Uhlenbeck process

TL;DR

This paper develops a large diffusion expansion for the transition kernel of the Lévy Ornstein-Uhlenbeck process, expressing the series in terms of generalized Feynman graphs and rules, addressing convergence issues.

Contribution

It introduces a novel large diffusion expansion for the transition kernel of the Lévy Ornstein-Uhlenbeck process and reformulates it using generalized Feynman graphs.

Findings

Derived a series expansion for the transition kernel

Re-expressed the series using Feynman graphs and rules

Addressed convergence issues in the series

Abstract

In this paper we study the L\'evy Ornstein- Uhlenbeck equation $\partial_t X_t=-m\,X_t+\eta$. The transition kernel of the L\'evy Ornstein- Uhlenbeck process is given by a series which is not convergent in general, a large diffusion expansion of the transition kernel of the L\'evy Ornstein- Uhlenbeck process will be given as well. The obtained series can be re-expressed in terms of generalized Feynman graphs and Feynman rules.