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

Transience and recurrence of a Brownian path with limited local time and its repulsion envelope