Reflecting Lévy processes and associated families of linear operators

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.