Brownian motion with drift ($X_t=W_t+\mu t$)

A Brownian motion with drift $\mu$ is $X_t=W_t+\mu t$ for a standard Brownian motion $W$.

For Brownian motion with drift $dX=dB+\mu dt$ in $(-b,b)$, the displayed probability of exiting at $-b$ follows by stopping the harmonic function $e^{-2\mu X}$. A unit-time increment larger than $2b$ has a uniform positive probability and forces exit, so the exit time has a geometric tail bound from a uniform escape probability and all polynomial moments. At zero drift the limiting probability is $(b-x)/(2b)$.

The laws on continuous paths of a standard Brownian motion and a Brownian motion with drift $\mu \ne 0$ are mutually singular measures on the whole half-line. The event ${\lim }_{t\to \infty }\omega (t)/t=0$ has full measure for the former and zero measure for the latter by the strong law for Brownian motion. On every fixed finite horizon, their laws are equivalent by the Girsanov theorem. The finite-horizon densities $e^{\mu B_t-{\mu }^{2}t/2}$ tend to zero almost surely while retaining expectation one, so they lack uniform integrability over all time.

For $a,b>0$,This follows from the Brownian reflection principle weighted by the Cameron-Martin theorem for a linear drift.

For $a,y>0$,

For $a,b>0$ and the convention $\sup \varnothing =0$,