On two-boundary first exit time of Gauss-diffusion processes: closed-form results and biological modeling

The Gauss-Diffusion (GD) processes and their First Exit Time (FET) through a couple of absorbing boundaries are here considered highlighting some specific relations. The corresponding linear stochastic differential equations are re-written specifying the coeffcient functions and giving them several theoretical meanings useful for biological modeling. The FET problem is considered in order to present and discuss a protein dynamics model based on Gauss-Diffusion processes in presence of two boundaries. Known closed forms of FET density are specialized for suitable GD processes and thresholds. In the context of biological modeling, relations between threshold values, mean behavior of the protein dynamics and input forces are given for the existence of a closed form result useful to describe the acto-myosin dynamics.