Abstract
This paper deals with the rate of convergence in $1$-Wasserstein distance of the marginal law of a Brownian motion with drift conditioned not to have reached $0$ towards the Yaglom limit of the process. In particular it is shown that, for a wide class of initial measures including probability measures with compact support, the Wasserstein distance decays asymptotically as $1/t$. Likewise, this speed of convergence is recovered for the convergence of marginal laws conditioned not to be absorbed up to a horizon time towards the Bessel-$3$ process, when the horizon time tends to infinity.
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