Abstract
In this paper we study the drifted Brownian meander that is a Brownian motion starting from u and subject to the condition that $$ \min _{ 0\le z \le t} B(z)> v $$ with $$ u > v $$. The limiting process for $$ u \downarrow v $$ is analysed, and the sufficient conditions for its construction are given. We also study the distribution of the maximum of the meander with drift and the related first-passage times. The representation of the meander endowed with a drift is provided and extends the well-known result of the driftless case. The last part concerns the drifted excursion process the distribution of which coincides with the driftless case.
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