  Probabilistic 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 the process taking place in a finite-time interval. The key issue is
to select the jointly continuous in all variables positive Feynman-Kac kernel,
appropriate for the phenomenological (physical) situation. We extend the
existing formulations of the problem to cases when the kernel is \it not \rm a
fundamental solution of a parabolic equation, and prove the existence of a
continuous Markov interpolation in this case. Next, we analyze the
compatibility of this stochastic evolution with the original parabolic
dynamics, while assumed to be governed by the temporally adjoint pair of
(parabolic) partial differential equations, and prove that the pertinent random
motion is a diffusion process. In particular, in conjunction with Born's
statistical interpretation postulate in quantum theory, we consider stochastic
processes which are compatible with the Schr\"{o}dinger picture quantum
evolution.
