Consider an ensemble of realizations of a stochastic process that all end at the same final point $x_f$ at time $t_f$! How can I calculate the probability distribution of points at which these sample paths were at an eralier time $t_i$ with? Or stated differently: How probable is it that the process, which is at $x_f$ at time $t_f$ arrived from an $x_i$ where it was at an earlier time $t_i$?
Please note that this question cannot be answered by using the backward kolmogorov equation. Also backward stochastic differential equations do not seem to work.
I cannot directly answer your question, but this paper seems to discuss the physics of Brownian particles subject to external forcing. Specifically see the Langevin and Fokker-Plank equations.