Schroedinger's interpolation problem and its probabilistic solutions

Probablistic solutions of the so called Schr\"{o}dinger boundary data problem provide for a unique Markovian interpolation between any two strictly positive probability densities designed to form the input-output statistics data for a certain dynamical process taking place in a finite-time interval. The key problem is to select the jointly continuous in all variables positive semigroup kernel, appropriate for the phenomenological (physical) situation.