# A stable Langevin model with diffusive-reflective boundary conditions

## TL;DR

This paper constructs and analyzes a one-dimensional stable Langevin process with reflective-diffusive boundary conditions, revealing different regimes, and extends to conditioned processes, persistence probabilities, and boundary problems.

## Contribution

It introduces a novel construction of a stable Langevin process with boundary conditions and extends existing results on conditioned processes and persistence asymptotics.

## Key findings

- Identification of two main regimes based on parameters
- Exact asymptotics for persistence probability
- Solution to the trace problem in the symmetric case

## Abstract

In this note, we consider the construction of a one-dimensional stable Langevin type process confined in the upper half-plane and submitted to reflective-diffusive boundary conditions whenever the particle position hits 0. We show that two main different regimes appear according to the values of the chosen parameters. We then use this study to construct the law of a (free) stable Langevin process conditioned to stay positive, thus extending earlier works on integrated Brownian motion. This construction further allows to obtain the exact asymptotics of the persistence probability of the integrated stable L{\'e}vy process. In addition, the paper is concluded by solving the associated trace problem in the symmetric case.