Natural boundaries for the Smoluchowski equation and affiliated diffusion processes.

Abstract

The Schr\"odinger problem of deducing the microscopic dynamics from the input-output statistics data is known to admit a solution in terms of Markov diffusion processes. The uniqueness of the solution is found to be 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 diffusion processes 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.