## Abstract

The paper is concerned with special one-dimensional Markov processes, which are Lévy processes defined on a finite interval and reflected from the boundary points of the interval. It is shown that in this setting, in addition to the standard semigroup of operators generated by the Markov process, there also appears the family of “boundary” random operators that send functions defined on the boundary of the interval to elements of the space L^{2} on the entire interval. In the case when the original process is a Wiener process, we show that these operators can be expressed in terms of the local time of the process on the boundary of the interval.

Original language | English |
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Pages (from-to) | 335-354 |

Number of pages | 20 |

Journal | Theory of Probability and its Applications |

Volume | 64 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 2019 |

## Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Keywords

- Initial boundary value problem
- Limit theorem
- Local time
- Random process
- random process
- BOUNDARY VALUE-PROBLEMS
- DOMAIN PROBABILISTIC REPRESENTATIONS
- initial boundary value problem
- limit theorem
- local time