In this paper, we formulate a computational framework for the assessment of pore diffusivity in a simple micro-channel. Diffusion under such conditions is of interest to both biological and physico-chemical applications that deal with the transport of particles in a rarefied fluid (where particle and molecular length scales are comparable), such as particulate matter removal in exhaust gas, diffusion of lipids through bio-membranes etc. The mobility of these particles may be influenced by neighboring particles and walls, as well as history effects in the surrounding flow. When a particle is not small in relation to the pore, these effects can no longer be accounted for via a point-particle method. Here, we treat the stochastic transport of particles in a bounded micro-channel within a continuum framework using a multiphase direct numerical simulation technique, the immersed boundary method. We show how this continuum method can be used to resolve the meandering path followed by a small particle subjected to Brownian motion, and present detailed investigations of the influence of the particle on the surrounding fluid flow field. Finally, we also discuss the advantages, challenges and implications of resolving a molecular phenomenon with a continuum technique.