On weak convergence of conditional survival measure of one-dimensional Brownian motion with a drift. (English) Zbl 0822.60094

Summary: 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”.


60K40 Other physical applications of random processes
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
60J65 Brownian motion
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