Forward Brownian Motion

TL;DR

This paper investigates processes that resemble standard Brownian motion in the forward direction but may differ in the backward direction, analyzing their growth rates and conditions under which they become two-sided Brownian motions.

Contribution

It introduces a class of processes with forward Brownian motion distribution starting from random points and explores their properties in the backward direction.

Findings

Such processes do not necessarily have the same distribution backward in time.

The maximum and minimum growth rates in the backward direction are characterized.

Conditions under which these processes are two-sided Brownian motions are identified.

Abstract

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.