Splitting and Breaking Brownian Pathways: Conditional Processes

In this chapter, we have calculated the fundamental physical properties of conditional probabilities when particular pathways are taken by Brownian particles as well as their mathematical representation. To gain new insights into the escape dynamics, we analyze the “fine structure” of these trajectories. Specifically, we divide trajectories into two segments: a looping segment, when a particle unsuccessfully tries to escape returning to the trap bottom again and again, and a direct-transit segment, when it finally escapes moving without touching the bottom. Analytical expressions are derived for the Laplace transforms of the probability densities of the duration of the two segments.