A "forward Brownian motion" is a process which has the distribution of standard Brownian motion (in the forward direction of time) starting from random points on the trajectory which accumulate at $-\infty$. 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. I will discuss the maximum and minimum rates of growth for these processes in the backward direction. I will also address the question of which extra assumptions make one of these processes a two-sided Brownian motion.
Joint work with Michael Scheutzow.