We consider one dimensional Markov processes of a special type which are the Levy processes with reflection valued on a finite interval. We show that in this case along with a standard semigroup generated by the Markov process there arises a family of "boundary" random operator mapping functions defined on the boundary of the interval to <nobr>$L_2$</nobr>- functions defined on the whole interval. If the original process is the Wiener process these operator are presented via a local time that the process spends at the interval boundary.