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Probabilistic solutions of the so called Schr\"{o}dinger boundary dataproblem provide for a unique Markovian interpolation between any two strictlypositive probability densities designed to form the input-output statisticsdata for the process taking place in a finite-time interval. The key issue isto select the jointly continuous in all variables positive Feynman-Kac kernel,appropriate for the phenomenological (physical) situation. We extend theexisting formulations of the problem to cases when the kernel is \it not \rm afundamental solution of a parabolic equation, and prove the existence of acontinuous Markov interpolation in this case. Next, we analyze thecompatibility of this stochastic evolution with the original parabolicdynamics, while assumed to be governed by the temporally adjoint pair of(parabolic) partial differential equations, and prove that the pertinent randommotion is a diffusion process. In particular, in conjunction with Born'sstatistical interpretation postulate in quantum theory, we consider stochasticprocesses which are compatible with the Schr\"{o}dinger picture quantumevolution.Comment: Latex file, J. Math. Phys., accepted for publicatio

Feynman–Kac kernels in Markovian representations of the Schrödinger interpolating dynamics