Brownian motion between two random trajectories

Consider the first exit time of one-dimensional Brownian motion from a random passageway. We discuss a Brownian motion with two time-dependent random boundaries in quenched sense. Showing $-t^{-1}\ln\bfP^x(\forall_{s\in[0,t]}a+\beta W_s\leq B_s\leq b+\beta W_s|W)$ converges to a finite positive constant $\gamma(b-a,\beta)$ almost surely if $a<x<b$ and $\{W_s\}_{s\geq 0}$ is another one-dimensional Brownian motion independent of $\{B_s\}_{s\geq 0}, B_0=x, W_0=0.$ $P(\cdot|W)$ is the probability condition on a realization of $\{W_s\}_{s\geq 0}$. Some properties of binary function $\gamma(b-a,\beta)$ are also found.