  We study discrete-time stochastic processes $(X_t)$ on $[0,\infty)$ with
asymptotically zero mean drifts. Specifically, we consider the critical
(Lamperti-type) situation in which the mean drift at $x$ is about $c/x$. Our
focus is the recurrent case (when $c$ is not too large). We give sharp
asymptotics for various functionals associated with the process and its
excursions, including results on maxima and return times. These results include
improvements on existing results in the literature in several respects, and
also include new results on excursion sums and additive functionals of the form
$\sum_{s \leq t} X_s^\alpha$, $\alpha >0$. We make minimal moments assumptions
on the increments of the process. Recently there has been renewed interest in
Lamperti-type process in the context of random polymers and interfaces,
particularly nearest-neighbour random walks on the integers; some of our
results are new even in that setting. We give applications of our results to
processes on the whole of $\R$ and to a class of multidimensional `centrally
biased' random walks on $\R^d$; we also apply our results to the simple
harmonic urn, allowing us to sharpen existing results and to verify a
conjecture of Crane et al.
