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

TL;DR

This paper studies the behavior of Brownian motion under local time constraints, describing limiting processes and proving conjectures for specific growth functions, with broader implications for Levy processes and local time conditions.

Contribution

It characterizes the limiting behavior of Brownian motion conditioned on local time growth constraints and proves two conjectures for critical growth functions.

Findings

Describes the limiting process when $f(t)/t^{3/2}$ is integrable.

Proves two conjectures for functions where $f(t)/t^{3/2}$ just fails to be integrable.

Provides a general methodology based on subordinator probability asymptotics.

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.