  By departing from the previous attempt (Phys. Rev. {\bf E 51}, 4114, (1995))
we give a detailed construction of conditional and perturbed Markov processes,
under the assumption that the Cauchy law of probability replaces the Gaussian
law (appropriate for the Wiener process) as the model of primordial noise. All
considered processes are regarded as probabilistic solutions of the so-called
Schr\"{o}dinger interpolation problem, whose validity is thus extended to the
jump-type processes and their step process approximants.
