Boundary problems for diffusion in a fluctuating potential

Using a newly developed formalism adapted for diffusion in the presence of (partially) absorbing boundaries, we treat the overdamped motion of a particle in a fluctuating potential and in a restricted space. The potential is assumed to be linear, with a slope switching between two possible values. The switching process is taken as the standard Markov alternating process. We first construct the free-dynamics propagator (no boundaries) which controls the evolution of the joint probability for both the particle’s position and the state of the potential. Secondly, at a given spatial point, we affix a two-channel boundary: a given channel absorbs the trajectories arriving at the boundary with the potential in a given state. We demonstrate the asymmetry of the filling process for the individual channels and we discuss their asymptotic occupation, depending on the mean switching frequency. Finally, we study the motion in a domain restricted by two boundaries of the above two-channel type and the corresponding first passage time. The simple problem-setting allows for an explicit analysis of the correlation between the particle’s motion and the potential-switching process.