Transience and recurrence of a Brownian path with limited local time

Abstract

In this paper, we study the behavior of Brownian motion conditioned on the event that its local time at zero stays below a given increasing function $f$ up to time $T$. For a class of nonincreasing functions $f$, we show that the conditioned process converges, as $T\rightarrow\infty$, to a limit process and we derive necessary and sufficient conditions for the limit to be transient. In the transient case, the limit process is described explicitly, and in the recurrent case we quantify the entropic repulsion phenomenon by describing the repulsion envelope, stating how much slower than $f$ the local time of the process grows as a result of the conditioning. The methodology is based on a fine analysis of the subordinator given by the inverse local time of the Brownian motion. We describe the probability of general subordinator to stay above a given curve up to time $T$ via the solution of a general ordinary linear differential equation. For the specific case of the inverse local time of the Brownian motion, we explicitly and precisely compute the asymptotics of this probability for a large class of functions.