A note on a deterministic property to obtain the long run behavior of the range of a stochastic process

Maher Boudabra, Binghao Wu

A Brownian motion with drift is simply a process $V_t^{\eta }$ of the form $V_t^{\eta }=B_t+\eta t$ where $B_t$ is a standard Brownian motion and $\eta >0$ \footnote{The case $\eta <0$ is deducible by…