Relationships involving spatial transitions for Brownian particles within a potential-well.

Using an optical tweezer apparatus we have trapped single latex spheres and analyzed their Brownian motion within a potential well. By considering transitions from various initial and final positions within the well, we experimentally show that the ratio of conditional probabilities, $P(x_f,t+\Delta t|x_i,t)/P(x_i,t+\Delta t|x_f,t)$, is independent of $\Delta t$. We also show the instanton times corresponding to last-touch-first-touch (LTFT) trajectories obey the equality, LTFT($x_i\rightarrow x_f$)=LTFT($x_f\rightarrow x_i$), shown by Bier et al. [Phys. Rev. E \bf{59}\rm, 6422 (1999)].