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We study random walks on $\mathbb{Z}$ which have a linear (or almost linear)drift towards 0 in a range around 0. This drift leads to a metastable Gaussiandistribution centered at zero. We give specific, fast growing, time windowswhere we can explicitely bound the distance of the distribution of the walk toan appropriate Gaussian. In this way we give a solid theoretical foundation tothe notion of metastability. We show that the supercritical contact process onthe complete graph has a drift towards its equilibrium point which is locallylinear and that our results for random walks apply. This leads to theconclusion that the infected fraction of the population in metastability (whenproperly scaled) converges in distribution to a Gaussian, uniformly for alltimes in a fast growing interval.

Random walks on $\mathbb{Z}$ with metastable Gaussian distribution  caused by linear drift with application to the contact process on the  complete graph