  The Schr\"{o}dinger problem of deducing the microscopic dynamics from the
input-output statistics data is known to admit a solution in terms of Markov
diffusions. The uniqueness of solution is found linked to the natural
boundaries respected by the underlying random motion. By choosing a reference
Smoluchowski diffusion process, we automatically fix the Feynman-Kac potential
and the field of local accelerations it induces. We generate the family of
affiliated diffusions with the same local dynamics, but different inaccessible
boundaries on finite, semi-infinite and infinite domains. For each diffusion
process a unique Feynman-Kac kernel is obtained by the constrained (Dirichlet
boundary data) Wiener path integration.As a by-product of the discussion, we
give an overview of the problem of inaccessible boundaries for the diffusion
and bring together (sometimes viewed from unexpected angles) results which are
little known, and dispersed in publications from scarcely communicating areas
of mathematics and physics.
