On Weak Convergence of Conditional Survival Measure of One-Dimensional Brownian Motion with a Drift

We consider a one-dimensional Brownian motion with a constant drift, moving among Poissonian obstacles. In the case where the drift is below some critical value we characterize the limiting distribution of the process under the conditional probability measure that the particle has survived up to time t. Unlike the situation where the drift equals zero, we show in particular that in the presence of a constant drift, the process in natural scale feels the "boundary."