  We consider processes which have the distribution of standard Brownian motion
(in the forward direction of time) starting from random points on the
trajectory which accumulate at $-\infty$. We show that these processes do not
have to have the distribution of standard Brownian motion in the backward
direction of time, no matter which random time we take as the origin. We study
the maximum and minimum rates of growth for these processes in the backward
direction. We also address the question of which extra assumptions make one of
these processes a two-sided Brownian motion.
