Transience and recurrence of a Brownian path with limited local time and
its repulsion envelope
release_epv2tahwl5bclc75dybjybbyei
[as of editgroup_4cl2ufvmnjbl5eq4drkjfkf56i]
by
Martin Kolb, Mladen Savov
2013
Abstract
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.
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1312.4131v1
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