  In this note we investigate the behaviour of Brownian motion conditioned on a
growth constraint of its local time which has been previously investigated by
Berestycki and Benjamini. For a class of non-decreasing positive functions
$f(t); t>0$, we consider the Wiener measure under the condition that the
Brownian local time is dominated by the function f up to time T. In the case
where $f(t)/t^{3/2}$ is integrable we describe the limiting process as T goes
to infinity. Moreover, we prove two conjectures in [BB10] in the case for a
class of functions f, for which $f(t)/t^{3/2}$ just fails to be integrable. Our
methodology is more general as it relies on the study of the asymptotic of the
probability of subordinators to stay above a given curve. Immediately or with
adaptations one can study questions like the Brownian motioned conditioned on a
growth constraint of its local time at the maximum or more generally a Levy
process conditioned on a growth constraint of its local time at the maximum or
at zero. We discuss briefly the former.
